rome or bust

Yes, the last thing you want in Rome is to rent a car.
Rick's guidebook to Rome makes many recommendations, more than we can here: do you have that book? It will be a low cost instruction book for your trip.
"Reasonably priced" is a relative term--this is one of the world's great touristic destinations, so brace yourself for some higher prices. You will want lodging with AC, even though early June isn't the worst of the Rome heat.

Use a search engine such as Trivago, Booking, etc. even TripAdvisor. You can search filtering all the amenities you need for the price you can afford.
As mentioned by Kent, reasonable price is a relative term. I like the level of quality that a 3 star hotel offers. In the historical center of Rome that should come in the neighborhood of 150€ ($170) a night, tax included, and sometimes even less. But some people will not settle for anything less than a 5 star resort, therefore, if you are one of those, be ready to pay more.
In my opinion the quaintest areas are either the historical center (Pantheon, Navona, Spagna, Campo de Fiori, Trevi) or Trastevere (beyond the Tiber River) which is a little away from the sights but a bit cheaper and very quaint. Closer to the main train station (Termini) or near the Colosseum (Monti) it is a bit cheaper, but not my personal favorite.

dr.bob,

I prefer to stay in the vicinity of Termini station as there are some nice hotels there and it's also very convenient in terms of transportation. From there it's easy to connect to Metro, buses or trains. I normally stay at either Hotel Sonya or Hotel Aberdeen, both an easy walk from Termini. If you have a Library nearby, check out the RS Italy guidebook as that has listings for good hotels in various parts of Rome and in various price ranges.

For travel in May, it would be a good idea to get something booked soon.

Rather than chasing down recommendations that you may find are already booked, I would start with booking.com to see what is available on your selected dates. You can sort by price, ratings, area. Remember that if the listing does not specifically say they have elevators or air-conditioning, they probably don't.

I liked 1 of Rick's recommendations, Hotel Portoghesi.. Less than 5minutes to Piazza Navona, and walking distance to all of the big sites. I can recommend this hotel....
We also stayed 1 night at the Raddison near Termini. The hotel was nice, but the area was sketchy. Lastly, we also stayed The Shire Hotel near the Villa Borghesi. This too was nice, but it is a high elevation and not too close to the big sites, so each evening, after walking all day, we had to walk uphill to our hotel!!!

We found our apartment on VRBO. It was around the corner from the Pantheon, walking distance to everything. We didn't have a car. We prefer apartments as you can spread out and relax, buy food from local grocery store, etc.