Rome where to stay.

Staying in the city center is what I have always done. Being able to walk the streets very late/early is a bonus too. I have stayed in the Piazza Navona area during my last 3 or 4 stays. A couple of times we stayed in the Via Veneto area, which wasn't too bad either. We have stayed in December and March mostly, I prefer the days to be cooler if we are walking around outside extensively. I have always been able to fine nice places to stay for around $200 per night. Good luck with whatever you decide.

We stayed in Hotel Aberdeen and it was great and not overly expensive.

What you save in money you will pay for with your time and transportation costs. We lived in Parioli, a nice Roman neighborhood Northwest of Villa Borghese. By public transport it would easily take 45 minutes to a hour to get the the historical center where you will no doubt spend most of your time. But it is a nice are with a local feel. The Hilton Garden Inn on Viale Liegi is well-positioned but I see it is over $300\night in September.

If you truly stay outside of Rome (Castle Gandolfo, Ostia, Ciampino, Frascati come to mind) the time getting into town is worth keeping in mind.

By staying in the center you can most easily go back to your lodging midday when you need a break.

Have you tried Booking.com to comb through budget possibilities?

Also, have you considered staying in a monastery? Lots of interesting reviews on the Forum (look on main page at top, use search bar). There seem to be some simple, inexpensive places to stay in central Rome. I would agree with others that staying outside Rome will increase your transport, hassle & cost. See what you can find in the center! https://www.monasteries.com/en-GB/search/accommodation/in/rome?rooms=1&adults=1&children&curr=USD

I totally agree that you would be better served by staying closer in the city where you can use your time enjoying the sites rather than making your way to them. A few years ago we stayed in the Trastevere area and found it a fun place to stay. It is an easy walk to the Vatican cross the Tiber. We truly felt that we were part of the local scene in this quaint section of Rome. We have stayed in other parts of Rome but found this to be a favorite.

We recently stayed in the Crown Plaza Resort which was maybe 5 miles NW of the Vatican. They have a shuttle that drops you off close to the Vatican, but I don't recommend them. Taking cabs back and forth to the center city gets a little expensive, and their restaurant was just too expensive.

I usually stay in the San Pietro/Ottaviano area which is a short walk east of the Vatican Museum. The A red line subway has a station there which means you can get there from the Termini. There are many bed and breakfasts in the area.

I also like the Pantheon area to stay in.

I often go to Google Maps and center in on a neighborhood. Then I hit the Hotels block at the top of the map, and a bunch of icons pop up with the price. Click on an icon to see all about the property, etc. You can also read what other customers have said about the property. I've never been disappointed by using this method to find accommodations.

I don't have ideas for outside the city, but we stayed at an apartment/hotel called Incentrum. An apartment/hotel can save money vs. a regular hotel, because you can have some meals (e.g. breakfasts) in. You also have more space.

They have a couple of locations (the one where we stayed was close to Piazza di Popolo and not far from the Spanish Steps). I think they have different suite sizes, too, some of which would be suitable for a family. (Be aware that there are no elevators, if stairs might be a problem for you.)

https://incentrum.it/en/home-page/

I second the option of looking at the area near the Vatican. We have stayed in the Prati area twice and really like it. You can still walk or transit to various sights and it is filled with lots of good restaurant options.

Near the Pantheon.

What is your budget?