Hotels in Rome

Look at the Centro Storico. That is the area that stretches from Campo dei Fiori to Piazza di Spagna (Spanish steps), passing through Piazza Navona, Pantheon, Trevi Fountain.
The Pantheon is probably the center of it all. Look at Booking.com and filter for the Centro Storico (Historical Center).

hey hey LADTM
when is this trip, how many days/nights, how many people is "we", "budget matters" means how much? prices have skyrocketed. we stayed in the piazza navona are which was great and near things we saw but you do walk to other activities/attractions, your decision. if your feet or your body aches/sore take a taxi to your place from a taxi stand, no flagging them down. keep address with you at all times
rome is always busy and crowded with many wanting the same thing as you are hoping and wanting. what amenities do you need: A/C, elevator/lift, climbing stairs & steps with luggage, read reviews and cancellation policies, checkin time 3-4pm and checkout time 10-11am, are you flying R/T to and from rome, any other places you hope to visit
booking.com
put dates in, what you want, look what's available on map, filters on left side to check. see anything email or call (hour difference) to see if you get better rate. keep all emails.
all the attractions you want to see, reserve ahead of time with direct site, many are date/time stamped. don't want till last minute since reservations go fast.
when you see something come back and ask questions for good bad and ugly. posters are here to give you info to help you have a great time with no stress. good luck enjoy
aloha

The area around the Pantheon is what I consider to be the focal point. But it’s also very busy with tourists. I stayed at Hotel City Palazzo Dei Cardinali last year, and I have it booked again for this year. I liked the location a lot to be convenient but also quiet. It’s very handy to take the metro at Roma Termini train station over to the Barberini metro stop which is near the hotel. (That metro will also take you to the Vatican.). It’s an easy walk to the Spanish Steps & Trevi fountain.

The Rome part of our trip will be around the end of September, maybe early October. There's two of us - my wife and I. As for price, I don't really know... $400 a night? We'll be there for 5 nights; I'd love to keep it around $2,000. Is that realistic? I know Rome is expensive. As for amenities, a lift is a big plus above the second (European) floor - thank goodness we pack really light, but there are limits!. Breakfast would be nice, but not mandatory. Other than that, location and friendly staff will win out. We don't need a spa or luxury toiletries or a "complimentary" bottle of wine. And a real door on the bathroom - what's with the glass doors and windows into the bedroom these days? LOL

@Jean: I just checked the Hotel City Palazzo Dei Cardinali website. It looks really comfortable and well-located. And about $1800 for 5 nights at the end of September - nice! Thanks for that.

$400 a night will certainly get you at least a 3 star hotel, which is more than adequate for most mortals (certainly for me and my wife). You might be able to snatch also some even more luxurious 4 star hotels for about $400 a night.

Check on Booking.com. Filter for these neighborhoods of the historical center:
Pantheon
Navona
Trevi

Piazza di Spagna is likely to be a bit more expensive than the others.

When using the Search function you need to filter by Type and Date

Here is search for alRome hotels for past year
https://search.ricksteves.com/?button=&date_range=1y&filter=Travel+Forum&query=Rome+hotels+

We prefer centro storico as Roberto has described
From there you can walk to almost anywhere in Rome

Booking.com is a good place to start your search
Use the filters for location, budget , amenities
Read the reviews there as those can only be left by people who have actually stayed at the property

A forum favorite is Hotel Smeraldo

Yes to Hotel Smeraldo. We loved the hotel…friendly, helpful staff, good breakfast, as well as convenient location (of the 3 hotels we’ve stayed in Rome, this was our favorite).There are lots of places to eat within walking distance, including a couple of great gelato shops.
We walked to the Vatican, the Pantheon, and the Coliseum and Forum. You didn’t mention the Trastevere, but we walked there to join a Food Tour, which I would highly recommend.

We always stay at Hotel Museum. It’s very near the Vatican Museum entrance. The neighborhood is fine with many “local “ smaller restaurants, grocery stores and gelato of course. The Hotel has breakfast on the rooftop terrace. We’ve been staying there since 2002.