If you contact Amtrak, they should have information on their best deals. In addition to the lines up and down the east & west coasts and the two or three transcontinental routes, there a few special luxury options. For one, there's something called the "America Orient Express" following a scenic route through the Rockies. This would not be so much a way to get from point "A" to point "B" as an event or "destination" all its own.

There are very few passenger lines any more, so you may not be able to reach your desired destination(s) by train in any event. And Maryrrf is right, the locations of many urban railroad stations are in neighborhoods that have long since been left behind (just as train travel itself has been left behind). However, in the very biggest cities at the beginning and end of each major route, most of the stations are still landmarks and in decent areas.

Bus stations, by the way, are even more likely than train stations to be in scary parts of town.

Renting a car and driving is a good idea, as long as you're not too nervous about learning to drive on the opposite side of the road. Our gasoline prices have increased drastically of late, but they're still lower than anywhere in Europe. Plus, you're free to see the countryside (more or less) from the interstate, or from the older highways that pass through towns, etc., at your discretion. Since the US is essentially a nation of motorists, your widest variety of options for meals, lodging, etc., are along the highways, not the railroads.

I'm afraid train travel is not as attractive as it was a generation ago, and is undoubtedly a less viable option in the US than in Europe. If your only travel needs are to move among the few cities with regular Amtrak service, you can probably find something to your liking -- but if you need more flexible travel options, the train won't do it for you.