Oh, geez. That's a HUGE question! This country is so huge and so varied that it's virtually impossible to name only a few sights worth seeing!
As far as cities go, my top list would be NYC, DC, BOS, CHI, SFO, and LAX, but there are worthwhile things to see in countless cities across the country, from PDX and SEA to LMY (oh, all right, Santa Fe--just trying to keep the 3-letter code thing going!) to STL to CVS to SAV.
And of course, there are a number of things in this country worth seeing that are more than just a single town: no one should die without having driven through the great Southwest, especially southern UT's incredible national parks. A drive up to Alaska (especially through Banff and Jasper NPs in Canada) is a must-do, too.
But that's a bit hard to do on Amtrak. So, to fit your criteria, what one destination would I pick? I'd probably do either WAS or NYC (especially New Year's Eve in NYC--now THAT would be fun!). You should be able to do WAS as a stopover on a trip from FL to wherever, since that's the primary point the CL and Cardinal connect to the Silver Service. You could do NYC as a stopover on a trip involving the LSL. BOS won't work as a stopover (unless you're heading to Maine!), but it's definitely worth a visit.
But once you get beyond those cities, how do you pick, say, the Mormon Temple Square over, say, the Gateway Arch, the Golden Gate Bridge, or the Sears Tower (though access to Giordano's pizza makes that one particularly compelling!)?
Other places, like L.A., are worthwhile not for a single landmark (who would travel all the way across the country just to see the Hollywood sign?) but more for the sum of everything available in the area--the good weather, the beaches, the sights and museums, and the glamorous lifestyle).
So it's hard to pinpoint just a single thing down. I'd stick with WAS and/or NYC for now and then decide to tackle the country region by region...when you're more specific about a region, we might be able to help you gather a list of the must-sees a bit easier!
