Definitely, as
Yuwipi Woman suggests, pick a region. Or, at the very least, spend half your time in one region and then fly to the other one.
You mention the trip from San Francisco to Chicago. If you're making journeys of that distance, do NOT take a train or the bus unless you have a specific love of that sort of travel, because that trip would eat at least two full days out of your schedule, whereas you can fly from SF to Chicago in under 4 hours (or about 6-7 if you have a stopover).
Also, if you plan ahead, air travel for that sort of journey isn't much more expensive than plane travel. I just looked at a couple of one-way flights from SF to Chicago, as well as a Greyhound ticket. The cheapest bus ticket was $88 (going up to about $160 if you purchase close to your travel date), and for that you have to sit on the bus for two days and 3 hours, and buy meals along the way. Or you can fly Southwest for about $160 and get there in half a day. Amtrak is $145, and takes even longer than the bus.
Getting around by Amtrak or Greyhuond over short distances is fine, although Amtrak is, as
D says, pretty expensive in the northeast corridor between Boston and Washington D.C. That's because demand on this route is high, with lots of business travelers, so Amtrak soaks them for as much money as possible. Amtrak, in the northeast corridor at least, is also more expensive on Fridays and Sundays, as those are the peak days for travel.
Still, if you need to go from New York to a city like Boston or Philly or Washington, i would recommend Amtrak above all other transport. You can, for example, get from NY to Washington for $72 per person. Also, because the train leaves from downtown Manhattan and arrives in downtown DC, you don't have to deal with the major hassle of dragging your ass to an airport all the way out in Queens (La Guardia) or Brooklyn (JFK). The NY-Washington trip takes just under 4 hours on the train, and the plane flight is just over an hour. But by the time you factor in getting to the airport, checking in, security lines, etc., it's cheaper
and quicker on most occasions to just use Amtrak.
If you really want to save money and take the bus, there are also alternatives to Greyhound. For travel into and out of New York, check out the so-called Chinatown buses (there's more than one company). I've used then quite a few times to go between New York and Baltimore, and they were always on-time, clean, and without any sketchy characters. The price is also right: it's generally $20 one-way, or $35 return, and only takes about a half-hour longer than the train, as long as you make sure that you don't time your arrival in New York for rush hour.
Overall, if you are comfortable driving on the "wrong" side of the road, and have the money to spring for a rental, i think the advice about driving given by other people is excellent. While train or bus are fine for getting you from one place to another, driving gives you much more flexibility. Also, as
Yuwipi Woman says, car rental is reasonably cheap, as long as you plan to bring the car back to the same location where you start. One-way rentals are much more expensive.
Another piece of advice: don't rent from airport locations, as they are often more expensive. And if you own a car in the UK, see if your own car insurance will cover you in a rental, because taking the rental company's insurance can get expensive. If you need to buy full coverage from the rental company, a $30-a-day car can quickly double in price. Also, the rental company's insurance is basically all profit for them, so if you take it, see if you can bargain with them for lower insurance rates. Read
this piece for some tips on US car rental.
