Hi Kathryn,
It's best to go on each direct website (and scroll through to end) to get accurate information/news as well as any changes during a particular season.
Colosseum -Roman Forum/Palatine Hill
https://www.coopculture.it/en/colosseo-e-shop.cfm < Timed entry as you choose your date and time.
No big bags - they prefer small to medium size.
Capitoline Museums
http://www.museicapitolini.org/
Borghese Galleria
http://galleriaborghese.beniculturali.it/it < Timed reservations are a Must.
No bags are allowed, so plan accordingly. Read their policies and "fine print."
Vatican Museum and SIstine Chapel < security check for the Vatican.
http://www.museivaticani.va/content/museivaticani/en.html
http://www.museivaticani.va/content/museivaticani/en/visita-i-musei/scegli-la-visita/musei-e-collezioni.html
Please be mindful of the Papal Audience on Wednesdays (St. Peter's Square - about 10 AM for 90 minutes or so)
St. Peter's Basilica
http://www.vatican.va/various/basiliche/san_pietro/index_it.htm
Free entry - no tickets are needed. You need to go through security unless you go straight from the Vatican Museums. If you have large bags or backpacks, you may need to check them. Checkroom is lower level - down by the stairs. Restrooms are there too.
Basilica di San Giovanni in Laterano
https://www.rome.net/basilica-st-john-lateran
Free entry - no tickets needed.
The Pantheon
http://www.polomusealelazio.beniculturali.it/index.php?it/232/pantheon
You didn't mention The Pantheon, but thought I'd list it in case. It's free entry so far. It's an active Church, so Mass is said. If interested, check their schedules. It's a gorgeous Church. No tourists allowed when Mass is in service.
I realize you don't want a guided tour, but to get some ideas and get a feel for Rome, please consider watching some of these https://www.youtube.com/results?search_query=The+Roman+Guy
Scroll until you see a video matching your interests. They have a lot of tips and good ideas to navigate Rome.
