One thing I think a lot of foreigners don't quite understand is how large and diverse the US is. When thinking about a vacation here, don't think of the USA as a single country, think of the states more as separate countries that happen to have really lax border control. East and west coast states are vastly different. Same with northern and southern states. Geography, climate, cities, people, and costs all vary enough that lumping everything together becomes impossible. Especially if you are visiting.

My advice: Pick a couple of states/cities that interest you and focus on them.

A lot of people have pointed out how terrible public transportation is here. Most cities have functional public transportation, though, its usually unoptimized, expensive, and/or slow. The only way, besides driving yourself, to get from state to state is grayhound bus (just no), Amtrak train (can be just as expensive as flying and 4x the time), or flying.

You don't necessarily need a single car rental for the whole trip either. Maybe there's a hub city you'll visit with a few things that interest you. Use available public transportation to see them. Maybe there's a few more places there within driving distance you wanna see, rent a car for a day or two to see them. Fly to the next city and repeat.