Ciao bella! Here you are in the Eternal City.
First thing's first. Get to the Vatican. Get your booty into St. Peter's Basilica (it's free) and go and see all of the magical things therein. You will be standing on top of the bones of St. Peter the Apostle, the hands that touched Jesus and the eyes that saw Him and the lips that spoke "You are the Christ, the Son of the Living God." Look especially for the Pieta statue of Mary holding the corpse of her son, done by Michaelangelo on the right side as you are walking in. The grave of JPII is right next to it. In the back, the major stuff around the altar was all done by Bernini, who is very famous in Rome.
After that, everything is basically a bonus. The Vatican Museums are just behind the basilica and they are worth a visit. Something like 10 euro but there's three miles of art from every major era in history and it ends in the Sistine Chapel. Another five euro let you climb to the top of St. Peter's Basilica and see the view from the top, which is beautiful and worth the 400 steps if you aren't too tired. It's much more than the heart of Rome; it's the heart of the whole Church, and thus, the whole world.
For the papal audience, 6am or even 7am should be a safe bet to get good seats, but double check what time the gates open. You will want to be there well beforehand, and then sprint after you get through security to the front. If you don't get the front row, don't worry; just focus on getting an aisle seat next to any of the wooden barricades and Papa Francis will drive around after the audience. Don't be fooled by the tiny nuns, by the way. They will look totally innocent and holy as they inch in front of you, but they aren't afraid to throw an elbow if it means getting to see the Pope, and then all of a sudden, their entire congregation will be in front of you and you'll be lucky to see the Pope's hat.
Hold your ground!
Bring whatever you would like blessed by the pope to the audience, but know that if you bring a large bag or backpack, that will have to go through the metal detectors, which will slow you down. It's best to carry nothing except a jacket, and make sure you dress well according to the weather! You will be out there for a while! After that, settle in, pray, and greet the people around you. Papa is very adorable, but often very stressed up close, and you should be able to get within a couple feet of him. Pro tip: if you sit somewhere near lots of babies and children, Papa is more likely to stop next to your section. Don't worry about a picture, there's plenty online, just take in the moment.
After hitting the essentials, make sure you plan a day for Assisi. You'll take a train out of Termini (which is both the main train and subway station) and you can be there for the whole day. They will drop you off at the bottom of the hill. Don't hang around there for too long; the only holy thing down there is the original tiny chapel that St. Francis restored. It's inside an enormous basilica that's to the left of the station, and then one block away from the hill (kinda tough to find if you don't know what to look for). Then, hop on one of the city's many busses (you might have to buy a ticket ahead of time at any of the magazine stands) and head up the hill. It should drop you off right next to a giant white church, which is where the bones of St. Francis are. The crypt is one of the most peaceful places in the world, even though it's full of pilgrims. After that, the rest of the city begins and you are free to wander around, check out the cool churches (there should be a free map online, or in a church brochure), and enjoy the city. The incorrupt body of St. Clare is chillin on the other side of the hill (although it kinda just looks waxy) and that chapel also has the cross which spoke to St. Francis, telling him to "rebuild my Church." Find somewhere nice to grab some pasta, pizza, or paninis (all Italian staples) and try and stay for sunset if you can. The view is stupendous. Nevertheless, be careful of getting back in time for your train, although I've found that most Italian conductors (who will come around to check your ticket) are generous if you happen to take a later train instead.
As far as Padre Pio, "San Salvatore in Lauro" is the name of the church with his relics in Rome. It's got his mantle and his gloves and several statues. Everybody in that church is absolutely pumped about Padre, and the atmosphere for mass was delightful. (I went on a Saturday afternoon, and they had a procession complete with a life-size statue of Padre, and then they took out his mantle after mass, and touched it to everyone's faces.)
For the rest of Rome, I'm struggling to tell you what to see if you only have a week. Much of it is amazing, and I don't want to overwhelm you. You can more or less walk the whole city, although if you're tired or headed to somewhere like St. Paul's Outside the Walls then you should take the subway. (Pro tip: the A line is awesome and smells like wet rocks, but the B line sucks and smells like dirty diapers covered in graffiti. Those are the only two lines, they meet at Termini, and they run like a cross through the city (unless the Italian government figured out how to run and the C line finally opened, but that's unlikely...)).
Overall layout: Rome is divided unofficially into several key neighborhoods, and more officially, into 7 hills. Each major area is marked with an obelisk (giant pointy statue), and you can usually see another obelisk every time you reach one, directing you through the city. They will lead you past the Spanish steps (towards the Trevi fountain, which is very tucked away) and onto St, Mary Major, which progresses to St. John Lateran, etc. etc... Basically, the Vatican is the only super cool thing on the other side of the river (the Tiber, or il Tevere), and the rest of Rome is on the other side. If you want a great view (prime for sunset) with lots of orange trees, check out Santa Sabina (on the Aventine hill). Get off on the Circo Massimo subway stop (B line) and walk up the hill. (Google maps works very well for figuring out the subway and walking combos, but you need a ticket to get on the subway. We just had a pass. The bus system is extensive, too, though.) If you keep going up the hill past the church, there's a keyhole in a large door from which you can see 3 different countries at once (Rome, Vatican City, and some embassy). It's pretty cool, but sort of unnecessary.
In that area, the Missionaries of Charity have a Rome convent. It's close to a church called San Giovanni e Paolo, which is also right next to a place where you could get relics if you wanted! There's a California priest who works with the missionary order set up next door, and if you ask, he'll bring you back and show you some first class relics, like St. Maria Goretti and St. Gemma Galgani, who are both connected with that religious order, which is why they have permission to "sell" the relics. (They cost money, but it's for the casings, not the relic itself.) If you go to that area, be sure to stop into the Missionaries of Charity convent. Just knock on the door, and they will take you in to see Mother Teresa's room in Rome, and offer to let you pray in their chapel (they mostly speak English). That could realistically all be a day trip, and you would finish with time left over.
On a bigger note, there are four major basilicas in Rome, and it's a common pilgrimage to visit all four. The first is St. Peter's, the next is St. John Lateran (which is actually the cathedral), then St. Paul outside the Wall, and then St. Mary Major. All are stunning and massive. St. John Lateran has incredible statues of all of the apostles (let me know when you spot St. Bartholomew ;)). St. Mary Major has a relic of the manger where Jesus was born (hard to be certain, but definitely a peaceful place). St. Paul outside the Wall has images of every Pope since Peter, and they are all worth seeing, but if it's too much, maybe just focus on the first two.
If you are tired, and just want to relax for a few days, then try wandering through some of the Roman parks and squares (called piazza's). Next to the Piazza de Popolo is the Villa Borghese (an enormous park), which I walked through everyday on my way to school, and in that is the Galeria Borghese, which is an art museum that houses several Caravaggio's and Bernini's. The sun set up (in the park overlooking the piazza) there is also amazing. In the Piazza de Popolo, there are three churches, and the one across from the other too (on the far side of the piazza) has a famous Caravaggio inside on the front left. The park in Trastevere (another neighborhood) is also wonderful. It's past the Vatican on the same side of the river, and that whole neighborhood is known for it's awesome atmosphere, abundant cafes, and great food. There's a number of cool churches there, but it's not even worth mentioning them particularly, because there are cool churches in every inch of that city and it would be best for you to wander into them yourself. Pro tip: keep an eye out for crypts because often they will have major saints hanging out below, usually the one that the church is named after. Otherwise, any of the major streets would be great to wander down. Another stupendous piazza is the Piazza Navona, and the pizza spots around there are great. Pro tip: restaurants on the piazza charge double, but if you wander down one of the side streets, you get the same great food for half price. Also in that area is the best gelato in Rome: the Frigidarium. Again, tough to find, but so worth it. If you make it there, get the Frigidarium flavor and then whatever else you want dunked in chocolate. Unfortunately, they're probably packed with tourists this time of year (my least favorite part of Rome), so walk down the side streets as much as you can. "Lungotevere" is the name of the street that runs along the river, and also the street that Bernardi is on, if you walk down far enough.
Speaking of places that have meaning for me in Rome, the church I love is Santa Caterina da Siena a Magnanapoli. It's immediately next to the Angelicum and has daily adoration. In the summertime, idk the schedule, and Fabrizzio probably doesn't wear his usual blue sweater when the weather is hot, but I was there virtually every day. (On a side note, if you actually want to visit St. Catherine's body, that is located in Santa Maria Sopra Minerva, which is kinda by the Piazza Navona.)
Ok. This is getting to be a lot, and I haven't even mentioned all of the places that I should yet. Realistically, Rome is wonderful and you will get the most out of your time there by making sure you see the Vatican (ideally, both during the day and at night), and then just wandering through neighborhoods and pooping into churches whenever you see them. (That's how Meg and I found the sight of a Marian apparition in Rome!) Probably just section off your days. Spend one day in the area around the Vatican, another in Assisi, another near Santa Sabina and the Missionaries of Charity, another near the Villa Borghese and Piazza de Popolo (probably walk down the main road (Via del Corso) to get to the Colosseum afterwards, and then one other day hanging out in the heart of the city, by the piazza Navona and the major basilicas. Like, grab a map at one of those tourist stands for a couple euro and just section off the main parts of the city and wander, planning your days around a couple major items. Drat, that plan misses a lot, but probably gives you a fair showing of most of Rome, assuming you try to get to Trastevere for dinner one night. Alas, a Google search is probably more helpful than me, because it's impartial and includes all the secular places that I ignored but are hugely significant like the Roman Forum and stuff.
Something like this would be perfect, because you can walk from
one side of the map to the other in about an hour, and it gives you a good
feel for the layout. Most maps will look like this and show you the
key places.
In summary, I think I at least mentioned all that I needed to in justice, but realistically, just enjoy it. You will undoubtedly pass 100 significant things without knowing what they are, and pay special attention to 100 insignificant things that look cool. There's random things that you will probably see just by wandering, like the ruins overrun with cats by the Gesu (OH, try and get to that church too! It's the home of the Jesuits, and sooooo beautiful! You can put that on your Piazza Navona day, if you foolishly try to follow my general plan).
Mostly, just buy gelato, and eat good pizza, and watch the sun set over the Vatican. That will be enough. Last pro tip (because I've devoted way too much time to this already): don't be fooled by a 6 euro margherita pizza. It will come out looking like an American, poor quality take-and-bake. Don't be fooled by the 11 euro one either; that's a tourist trap. Aim for 8-10 euro and ask (if you can) whether it's made fire-burning pizza oven (the mark of a true Italian pizza). There will also be many places that have pizza slices for 2 euro; personally, I wasn't a huge fan of the odd, rectangular pieces, but many of my classmates enjoyed them. If you see it, my absolute favorite meal was Bucatini al' Amatriciana, which is a wonderful pasta dish (with a glass of red wine), but don't forget to try Carbonara, too. If you're in the mood to be foodie, a google search will reveal the best restaurants for each, but you will probably be fine just wandering and finding a place that looks good.
I hope that helps, even if it was a lot, and I am so happy that you are in Rome. If you want more places, or a more condensed list, or have a question about anything, email me! We can set up a skype, too, if that would work better. Great!! I hope you get this before you wake up, maybe spend your first day around the piazza navona and the center of the city, since you are going to St. Peter's on Wednesday. I love you and I'm praying for you!
Ponder love today.