Depending on the time of year, you might be able to fly for less than going by Amtrak.
I have a trip planned for July 2013 at Bonnet Creek and it is for 4 adults and 2 kids. I just priced Amtrak out of Penn Station for 4 adults and 2 kids and the price for Amtrak is more than we are paying by flying on JetBlue.
Last summer I went by Amtrak out of Penn Station to Williamsburg, Virginia because I couldn't find affordable nonstop flights. The price for Amtrak was great, but not our trip. It was supposed to be 9 hours and ended up 11 hours. The youngest was 2 at the time, and he was either crying or running around. People got annoyed and started complaining.
We took a late night train thinking that he would be tired and sleep through most of the trip, we were wrong about that. The other child was 12 at the time, so he had his DS game to entertain him. He was uncomfortable sitting for so long, but that wasn't too bad. We even packed a cooler bag and brought sandwiches and drinks for the kids to eat. I'm glad I did because it's expensive on the train. I don't think I would ever do that again.
Many, many years ago when I was in my 20's, I took my younger sister on a vacation and we went by Amtrak to Miami, Florida. What was supposed to be 31 hours ended up to be 36 hours or more. We ended up with no working toilets or water. I'm sure Amtrak has come a long way since then.
Good luck on your trip. Bring plenty of snacks and drinks for the kids. It is quite expensive on the train.