As an American I think you get a 3-month tourist visa automatically on entry but verify on the US state dept and french embassy sites.
pap.fr has lots of short-term apartment rentals. It should be easy to find a place using that site or others. pap-vacances is a section too that specializes in short-term rentals. You could alternatively do the hostel route. Less privacy but you meet lots of people that way.
Best cities? If you have 3 months, I would visit two or three. Paris is obviously the most expensive; housing in Paris always ranks as one of the top 5 most expensive places for housing and living in the world. Where else? Depends on your interests. If you like beaches, try for early spring and spend a month in cote d'azur. If you are going in winter and like skiing/snowboarding the somewhere in the alps like Chamonix. These are both more expensive, obviously.
Get a guidebook. Let's Go has up-to-date options for budget travel. On the other extreme, Michelin guide has the top French-vetted choices. Either way cheap places need lots of advance notice especially in peak seasons.
Or you could just do what I did which is fly by the seat of your pants. I rented a car and drove around , staying in cheap hotels. When I found a spot I liked, I looked for short-term lodging and stayed.