We go quite often. Since it's only an hour away we like to run up for lunch. The El Tovar is the original hotel and has a very nice dining room. Taking the train from Williams is something I've wanted to do, but it is quite pricey. There are quite a few nice motels in Tusayan, the little town right outside the entrance to the South Rim of the park. Also from Tusayan you can take helicopter or airplane tours of the canyon. Flights are not allowed below the rim for safety reasons. Enter the park and leave your car in one of the first parking areas and ride the bus if you are physically able to get on and off the bus. Buses are free and arrive/depart at each stop about every 15 minutes, so no worry about getting stranded anywhere. See the entire South Rim. There is nothing that is not worth seeing. There are LOTS of great photo spots. Walk as much of the rim as you are able. You can get off the bus at one stop, spend time looking and taking pictures, get on the next bus or walk to the next stop, etc. Be sure to take the bus to the Hermit's Rest and take advantage of the bus stops along the ride. We vacationed at the park a few times before we moved to northern AZ. We love every trip we take there. There is so-o-o-o much to see. Grand Canyon is beautiful in the snow too, and there aren't nearly as many visitors during the Winter. Try to avoid spring break (schools) as it is horribly crowded at that time. Here is a link to Grand Canyon National Park information.
http://www.nps.gov/grca/planyourvisit/index.htm
If you have plenty of time you can go the both South and North Rims, but it takes most of a day to drive from one to the other, and no mass transit that I know of. North Rim is less visited and much quieter. At 1000' higher elevation than South Rim, it is also cooler.
Don't be in a hurry. Take time to enjoy all the beauty and you will have a great trip.