I think this is a great idea. The easiest places to do this are definitely major cities. Bonus points if they have great public transportation. I would definitely research that as not all cities are created equally on the public transportation front. For instance Austin, TX is one of our favorite places to visit, but it has terrible public transportation. We had a hard time getting anywhere without resorting to taxis, which can be expensive. It is a fairly easy city to bike around if you're interested in that, though. (Also as a side note we haven't been there in a few years, so it might have gotten better...hopefully!) San Diego, CA, on the other hand, has great public transportation. For $5 a day or $15 for 4 days you can get unlimited transit passes which can get you almost anywhere you want to go. It's also easy to cart bikes around on the trolleys and buses.
You can do this in more rural areas, but it's going to be more challenging. DH and I like to travel to rural areas without cars and it can be difficult, but half of the fun for us is trying to figure out how the heck we're going to get anywhere. After having lived in rural Japan for a few years with little public transportation and no car for half the time we were there, it became a game for us.