My mother just took Amtrak from Boston to California this summer. She took an overnight bag (for 3 nights) and checked through another suitcase, just like on the airplanes. She never had to deal with it until the other end. I don't know if it was because she had a sleeping compartment but you could check. She got a lot of info from the Amtrak website but also had good conversations on the phone with the customer service reps. When I took her to the train, there were lots of families checking a lot of baggage through. Some of them were just going as far as Chicago, not going all the way through to California. I can't imagine there wouldn't be some sort of baggage process on your train as well.
My friends live in center city Phila and always lived in their strollers, taking the kids through all the narrow streets. They never used their car except to leave town for visits.
Any furniture rental place should have the items you need. You can google it, try for a chain like Taylor Rental, or have your friends do the work. They could pick you up at the station with the 2 car seats, unless you are taking a cab. You might want a car seat on the train for the baby so you don't have to hold him/her all the time. It will certainly help with napping. The train is good because there is the same scenery as when you drive (sometimes more interesting), and you can walk around while still making time. The problem is, you can't get off, and you have the same worries about a fussy kid bothering other travelers as you have on a plane. If you bring some activities for the 2 year old and hope for naps, and just don't worry about the other travelers, you'll be fine. Bring snacks too in case the toddler doesn't like what they have in the dining car. Again, the customer service people should be able to help you on that front. Amtrak is really pushing train travel and amenities because of the gas crunch, so there may be a lot of options for you. Also, Thanksgiving weekend is the heaviest travel weekend of the year, and trains will move when highways are bottlenecked!
Good luck and have a good trip!