Hello,
I have been to Rome twice so far, and I can comment with first-hand experience about the below.
- Vatican Museums, the Sistine Chapel, St. Peter's Basilica
If you are intending to visit The Vatican Museums first, you will go through a bag and security check.
You put your bag, jacket or coat and any hat you may be wearing in bins on the security belt. There are usually two guards.
A bag of 14- to 15- inches or under were accepted. No food. Water bottles were okay'd as long as they were put into your bag < or that is what I experienced.
It may be best to carry a softer day bag or softer backpack if you can. If it looks to be bulky or intrusive, the guards will have you check it.
If by chance, you or anyone in your party uses a mobility device, they were accepted after the security screening.
No large umbrellas or large backpacks or large bags in general allowed. The guards will direct you to "free check" them.
Keep in mind, if you check any large items, it will take time to retrieve the items. I don't know how crazy or crowded the free bag check will be in The Vatican.
(I visited The Vatican Museums once and visited St. Peter's many times during my trips. The security check was consistent.)
There is a large free bag check at St. Peter's. At the bottom of the stairs, you will see glass doors -- there will be the cube style holders for personal items. There are also restrooms down the hall.
No recording or camera devices allowed in The Sistine Chapel. No talking either.
If you are with an official tour group, after your Sistine Chapel visit, your guide will escort you through the back entrance which will come out to the terrace of St. Peter's Basilica entrance. At that point, you do not need to check your bag or personal items again. But, there are security guards dressed in suits eyeballing the visitors.
If you visit St. Peter's first, you will go through a similar security and bag check like The Vatican Museums. By the way, St. Peter's is free to enter even if you do not visit The Vatican Museums.
No recording or camera or talking allowed in The Chapel of The Blessed Sacrament. (in case you visit while in St. Peter's.) The Chapel is jaw dropping beautiful. Gorgeous altar. Ornate. You will just have to "snap a memory in your mind." lol
(I realize you did not ask, but here is some helpful info about navigating St. Peter's http://stpetersbasilica.info/index.htm)
Pretty much the same applies -- no big, bulky bags, backpacks, and luggage. Water bottles seem to be okay. I would just put it in your bag and let the guard decide about its future. I don't think there is a bag storage area at the Colosseum for bulky items.
You did not mention The Borghese Gallery -- No bags of any size allowed. That is where a money belt would come in handy for your utmost valuables.
In the wintertime, it may be different. A tiny cross body bag could be worn under a coat without it affecting any sculptures. Some people have reported they were given plastic carry bags to put their phones and "wallets" in. I do not know this for sure.
The Borghese Gallery's website states a no-bag allowed policy.