Calicash:
I will continue leaving out the option of driving, which if your parents had a reliable car, and were willing to do is a cost effective option for moving you and your stuff into the dorms.
However, As I mentioned Amtrak is very doable and can be very enjoyable if you look at the train ride as an adventure. You can take carry on luggage right into the train car with you but you can also take additional suitcases or boxes and have them checked on at Penn Station and taken directly off at Tuscaloosa. I have brought suitcases, coolers, shelving (properly wrapped) and had them checked in -see their website for info. Everything arrived safe and sound. I met interesting people, the train was clean and well air conditioned (take a small blanket) and like any mode of transport, remember to keep your valuables on your person. Your luggage will be taken off the train from the luggage car and brought right to the station house. You will need to carry off your own personal belongings, if you carry on items. Be prepared to do that. If you did need help, your conductor may be able to assist you to the door.
Be aware that in Tuscaloosa you dismount via stairs to the gound. In Penn station, you walk right onboard the train as onto the subway.
The cab ride from Tuscaloosa is about 5 minutes to the edge of campus and when I went the cost was about $8, remember to tip your driver (just like in NY).
This could well be your most economical way to initially come down to UA, if you book in advance, use a student discount card, and take extra luggage.
Flights if booked in advance can be a good price and if your parents take out a credit card with the airlines, your family can accumulate frequent flyer miles much more quickly and then you will probably have at least one or two free flights during your college years. Southwest often runs a promotion giving miles to join.
Join all the frequent flyer programs and book all your flights using your frequent flyer number in order to get your miles credited.
If you are flying, bringing a second piece of luggage can be very cost effective, even if you have to pay for that second checked bag. The cost of shipping a box from NY to Tuscaloosa will vary with the box size, but usually is about $1 a lb plus the cost of the box, packing and insurance. Boxes shipped on a Monday from NY will mostly arrive by Friday in Tuscaloosa. A 25lb box will cost approximately $40 - $50 depending on the physical dimensions of the box via UPS. Sometimes, it is cheaper to ship US Postal.
Note: SouthWest allows two free checked bags.