How can I travel between US towns without flying?

Car rental is the most flexible and enjoyable option.

Inter-city buses in the U.S. do not come close in comfort, frequency or convenience to ones in most of the world (e.g. Europe and East Asia). With few exceptions (5 or so major cities), once you get to your destination, public transport is sparse and infrequent, while walking the streets is unusual and at times unsafe. Buses are generally used by people who can't afford a car and must get somewhere, and not for tourism / pleasure trips.

Trains work well in established corridors (east coast between Boston/NYC/DC, and San Francisco Bay Area in the west), but otherwise can be both more expensive and less convenient than airplane.

Something nice about Amtrak is that you can cancel your tickets without penalty.
So you can always change your plans at the last minute.

Another option is to rent an RV. It's more expensive, of course, but it's very flexible and you save on hotels (plus you meet all sorts of weird and wonderful people at the stops):

If you are traveling between cities on the east coast, the so-called Chinatown bus lines are one of the cheapest option. They usually start and end in the respective China towns of New York, Boston, Washington DC, Baltimore, Philadelphia and other cities.

There are various different companies running these buses, you can search online or just ask around in China town, they have small offices from which the buses are leaving.

I recommend Bolt Bus first and MegaBus second for travel on the East Coast (Boston, New York, Phili, DC, Baltimore).

As mentioned in other answers, they are both generally cheaper than Greyhound/Peter Pan (in fact Wikipedia tells us that "BoltBus is ... a 50/50 venture between Greyhound Lines and Peter Pan Bus Lines"). Not only that, but they are often just as cheap if not cheaper than Chinatown lines if booked in advance.

You can board either one by just showing your reservation number on your phone, from the email or SMS they send you.

Their locations are convenient as well. In Boston and DC they leave from the same place as all major buses and trains (South Station, Union Station). In New York, Bolt and Mega leave from (in my opinion) more convenient locations than other buses. They so street side pickup in the 30's (Mega on 31/9th ave, Bolt on 33/7th or 34/8th depending on destination). Port Authority, the major bus terminal in New York, is an unpleasant place to be, and it is very annoying to navigate. Chinatown is more out of the way, although if you are going to be downtown anyway, that location is better.

Bolt Bus pro tip: register for an account before you buy. That way you get an "A" ticket, which means you get to board the bus first! Make sure you get a seat with a power outlet, since not all seats have one!

If you are travelling from the east coast to any other major city, in my opinion you should fly. One-way fares are often just 50% of RT. Amtrak is often (not always) more expensive than flying.

Amtrak > Buses IMO - but don't go to as many places.

If you're talking LONG distances - say Chicago to Seattle or similar, something that will take over a day however you travel (bus/train), then train will be far far far more comfortable, and may well be an enjoyable experience in its own right.

On long-distance lines, buying a sleeper is optional: the regular coach seats are spaced much farther apart as seats on an airplane, and they recline quite some ways - not close to flat, but definitely closer to 45' or so - and even have a food rest that pops up, like a La-Z-Boy. (The only awkward part is there's no armrest to separate your part of the seat from your neighbor's!)

You can get up, walk around, head to the viewing car, buy a bottle of beer or wine or snacks, and watch the scenery go by while chatting to other folks. (I think you can also pay to use the dining car for casual sit-down service, once the first class travelers are done using it - I did this on a 24ish hour trip from San Francisco to Seattle a couple of years back.)

You can pay extra for a sleeper cabin, but it's quite pricey - you're really paying for a room with a bunk bed, so it's bad value if travelling solo vs travelling with a friend you can split the cost with - but it basically gets you first class service: meals are then included, plus you get to use a fancier first-class viewing car. But if you view it more as a 'land cruise' than just transport, you might consider it worth it.

Do note that some Amtrak lines can be notoriously delayed; Amtrak owns the trains, but not the lines, and the freight companies that own them give their own freight trains priority, so it's not unusual for an Amtrak train to slow down and stop for some time as the freight train ahead clears a switch. The train from Los Angeles to Seattle is often known the "Cost Star-Late" rather than "Coast StarLight" for this reason...