The Statue of Liberty is a great tourist destination for lots of reasons. First, of course, is the statue itself, which really is a very impressive sight: it's massive and really rather beautiful, and the views of downtown from Liberty Island are lovely. Second is the fun of getting to the island: you have to take a ferry from Battery Park, which is always entertaining, though it can be extremely cold in winter. Third is the people-watching: real New Yorkers never go to places like the Statue of Liberty, unless they're escorting out-of-town guests, and as a result, the line for the ferry is even more delightfully international than most of New York.
Entrance to Liberty Island is free, but in order to get there, you must take a Circle Line ferry. Tickets cost $11.50 for adults, $4.50 for kids, and $9.50 for seniors. Ferry tickets can sell out, so it's a good idea to buy ahead, especially if you want to go inside the base of the statue--for that you'll need a separate free ticket that admits you on a guided tour. These are time-stamped so that you have reservations for a single tour. The time-stamped tour tickets sell out well before the ferry tickets do, so if entering the statue is important to you, plan ahead.
If you don't enter the statue, there isn't very much to do on the island except walk in a wide circle around the statue, admiring it and the views of Manhattan and New Jersey. But it's a very pleasant thing to do.
The ferry leaving Liberty Island takes you to Ellis Island, which is a much more interesting (if less picturesque) spot: the former immigration center that processed hundreds of thousands of immigrants in the early 20th century is now a museum documenting immigration to America. Most of the exhibits consist of photographs of immigrants, although there are also some cases full of objects people imported with them (embroidered dresses, silverware, books, etc.) and later donated to the museum. I found the photographic exhibits surprisingly moving: they're well selected, and the information provided along with the photographs is thorough and varied. The museum manages to do a good job of showing just what immigrants were up against in their home countries (the Irish potato famine, massive unemployment in Italy), and then what challenges they faced in America (anti-immigrant sentiment, the difficulties of assimilation).
If you're interested in doing genealogical research on Ellis Island, there's a library, although you should contact them in advance for permission.
There's food available everywhere on this trip: while you stand in line for the ferry, you'll be surrounded by pretzel carts, and there's a concession on the ferry, a fast-food restaurant on Liberty Island, and a cafe at Ellis Island.