Trastevere or just outside Vatican

You need to see if Trastevere has bus service into the center of Rome or to the sights you plan to visit. I do know that not far from the Vatican there is a metro stop and you can go to The Spanish Steps and to area by the Termini Station ( which has 4 churches in the area to visit) and the Colosseum ( all by metro) plus there is bus service near the Vatican. You will be using both buses, metro and lots of walking while in Rome. And you need to have access to both buses and metro. You may use buses more, but I like the using the metro. So you need to look at a map to plan out your days as to where and what you want to see and how to get to each site by public transportation. You can only get to the Pantheon area by bus or walk if you are nearby. There may be a lot more great restaurants in Trastevere. I have not stayed in either area, but have been to both areas and each has things to see. The only things I think to do in the Vatican area is see the Vatican ( can do that in a full day or as long as you want to stay at the Vatican during the day) and down the street ( good walk but not very far) is the Castel Sant'Angelo which is a good place to go for great views and a must if you are near the Vatican. Trastevere has some nice churches, great restaurants and more where people live. If you have not, do get the Rick Steves Rome book and look at the Rome week long tour for ideas and how they do the tour so that time is not wasted. I did the Rome tour twice and stayed in Rome three times near Baths of Dioceietian. Have a great trip! Do look at YouTube has that may help you get to know Rome better. Picture is worth a thousand words!

You can walk to the Vatican museums in 40 minutes from Trastevere and to the Colosseum in around 30minutes. Will also take you around 25 minutes to the Pantheon and Piazza Navona.

My vote is for Trastevere! I have stayed there each time I have visited Rome. It's a neighborhood just across the river from the Centro Storico, but you can walk to most places of interest from there. That is not the case for staying just outside of the Vatican. There is a handy bus from the neighborhood you can catch to the Vatican as well as to Testaccio (great food market). The neighborhood overall is well served by buses. Also there is a tram (#8) that takes you from central Trastevere to the Centro Storico. Honestly though we walked most places. The monastery is in a quieter part of the neighborhood too, which is a plus.

While you're there, do eat at Le Mani in Pasta, Via dei Genovesi 37 (reservations required), as well as La Gensola, Piazza della Gensola 15 (reservations also a plus).

I vote for Trastevere also. I have stayed there several times. There is the tram and also several buses (I think H and another) that are very convenient. There are lots of interesting areas to explore also.

Some places near the Vatican may be quieter, but Trastevere is my preference after the Centro Storico. Trams and several buses will get you to the other side of Rome.