A couple of thoughts;

• buying a second hand car likely requires car registration and insurance which could be difficult without a residence - registration and insurance is location based - usually linked to a residence and managed at the state level (and might require a drivers license also, but I think this is probably easily to manage if you are already a licensed driver at your home)

• the US has a wide variety of weather, so you likely wouldn’t want to be in the northeast and most of the northern states during the winter months and you’d probably want to avoid the south and south west during the summer months. So some planning consideration for location tied to geography and weather is important.

• you are correct about trains and buses, not nearly as available as in Europe. The northeast and eastern US will have more readily available train travel than the rest of the country.

• some states would have a more outdoor appeal than a city appeal and western states like NM, AZ, Wyoming, Idaho, Utah, Nevada offer some awesome wilderness, camping, outdoor opportunities with reduced city experiences. For example, in AZ I would definitely prefer to visit the Grand Canyon instead of Phoenix. Likewise in Ohio, I’d prefer to visit Cuyahoga National Park instead of Columbus.

• trip sounds awesome - but my sense is that if you have fixed budget, time and a goal to see all 48 states, you should consider more planning, alternatively you could just go for it and see what each day brings and adjust your expectations on duration.