Over several years from 2015 onwards, I and my daughters took 5 of the long-distance Amtrak trains from East Coast to West, and up the west coast from LA .to Seattle (and vice versa.) It was absolutely the highlight of my life, and we got to see so much of America! So many states and such incredibly varied scenery, from the vastness of the mid-west with its tiny towns and huge landscapes, to some of the big cities.
Funnily enough, we never got to New York and I don't regret that, but Boston, Chicago and San Francisco were all amazing in their different ways (we stayed overnight at each, with my favourite by far being San Francisco, such a pretty place with flowers and hills and views over the Bay and a real holiday feel with te Embarcadero and fish and chips on the front there (seagulls, tho!!!) and at the time it had a wonderful shopping mall (since sadly closed, alas) once you got away from the Bay area. (As others have said, I hear it has gone downhill since those days, which is a huge shame.)
I wouldn't call it a cheap holiday exactly, but the trains themselves are very good value. The California Zephyr went through some amazing landscapes (Rockies, Colorado river) and I think, cost about $400 for the entire journey of thousands of miles. Mind you, that was our first trip and we unwisely thought we could spend three nights in 'Coach' - they have pretty decent, spacious seats which recline, and blankets handed out etc. It was hard on our backs though, so we always had sleepers after that at considerably greater cost, but well worth it, and came with the bonus of included (excellent!) meals. I felt well looked after on those trains - each coach has its own attendant who fills up the coffee early in the morning and doesn't take any nonsense from passengers. BIG glass Observation car like a lounge, with swivelling seats and amazing views both sides, on every train.
Gosh, this is long, sorry! But as you can see from this map, if it posts readably, it's a cheapish and fun way to see as many states and cities and towns and landscapes as you can.