I second the Hudson Valley for hiking and you can get to a trail leading to the Appalachian Trail from Harriman Station (Main/Bergen/Port Jervis train line). I've seen quite a few people doing precisely that.
Another option, if you like museums - how about the local nation parks and historic sites? There are several in NYC that you don't need a car since you already have a train pass. Some of them are quite inexpensive, others not so much.
A partial listing - The Hamilton Grange, Federal Hall, Governor's Island, Statue of Liberty, Ellis Island, Roosevelt Home, Grant's Tomb, Castle Clinton. There may be one of two others that I missed. You may also want to check out the Tenement Museum as well (not a National Park and a bit more expensive).
Good luck and enjoy your vacation!