We've had a lot of "favorite train-watching spot" threads here in the past, and I've usually responded to each in the same way, but this didn't ask for favorites, just for the "best places to watch trains go by"--not necessarily to photograph them or anything special. So, with that in mind, I offer the following, not in any particular order.
--The UP between Gibbon and North Platte: sheer volume.
--Elmhurst, Illinois (UP): volume, variety (freights and scoots), amenities (food, hobby shop), and the home signal for a control point to observe.
--Anywhere along the BN(SF) triple-track raceway: volume, variety (freights, dinkies, Amtrak), preferably on a stretch straight enough that you can see how a block signal system works. LaGrange is probably the best place for all of this, because it offers food, a hobby shop, and the IHB a short distance east for added volume.
--Rochelle, Illinois (BNSF, UP): The crossing of two busy mainline railroads, a nice gift shop (any day but Monday and Tuesday), and a great gathering spot. You can get three main lines at Fostoria, Ohio (CSX, NS), but it lacks the centralized location for viewing more than two at a time, closely.
--Kirkwood, Missouri (UP): Pleasant surroundings, good volume, hobby shop nearby, and the ability to hear trains climbing grades in either direction.
--Horse Shoe Curve (NS): For a flatlander, the ability to see a long stretch of non-level railroad, and the struggle it takes to move the tonnage up (and down!) same. Volume isn't bad, either.
--North Canal Street, Chicago, during the rush hours: Two levels of Metra choreography, with a little Amtrak thrown in. CTA trains to boot!
--Blue Island or Dolton, Illinois (IHB, CSX, and friends): Volume, variety, and the chance to see how complicated some places can appear.
--Griffith, Indiana (EJ&E and CN): Nothing like it used to be, but it's interesting to see how many lines used to go through here. Probably even nicer when the "museum" is open.
--Durand, Michgan (CN, TSBY): A good place to see how important railroads were to some cities. An astounding station building for a town that size. It's kind of sad to visit for a person who still remembers what used to be there, but for a first-timer, the chance to use one's imagination makes it a worthwhile experience. Still quite a few trains, too.
Wow! I made it to ten, after all! I'm sure there are many other places, just as worthy or meaningful, but these can be enjoyed whether or not you bring a camera.