The Statue of Liberty was dedicated to the US from France in 1886. Its copper skin is 1/8 inch thick and has turned green through the years due to oxidation. In order to understand how large the Statue is, Lady Liberty's eyes are 2 feet in diameter and the book she carries is 2 stories tall.
There are a variety of ways you can see the Statue of Liberty. You can request a window seat on your flight and look at it from above. You can go to Battery Park and catch a glimpse of it from far away. You can take the Staten Island Ferry or you can do the Statue of Liberty tour.
Battery Park is easy enough to get to. We walked from our hotel (Cosmopolitan in Tribeca), but there are several train and bus options, for example N or R to Whitehall Street. Here you can walk along the Hudson River and see the Statue way out in the distance.
The Staten Island Ferry brings you a lot closer to the Statue and is completely free. The one-way trip is about 25 minutes. You pass right in front of the statue and can take great pictures of both the statue and downtown Manhattan. It's comfortable, but it is not a luxury by any means. You can stand outside or sit inside. There are many locals, but there is also a fair amount of "tourists" as well.
You can get here by taking many trains, for example the 1, to the southernmost point.
Statue tours. It is free to visit the Statue and Ellis Island museums. But, you do have to pay to take the ferry to get to these locations. The ferry ride is $11.50 adult /$9.50 seniors / $4.50 children. It leaves Battery Park in NY and Liberty State Park in NK. When you arrive on liberty Island there's a gift shop and a snack shop. You can purchase an audio tour of the island. Unless you've pre-purchased an in-statue tour, at this point you really only have the option of walking around the statue. They no longer let people go in the statue. It's a nice walk (if the weather is good) and it is very relaxing. Onto Ellis Island Immigration museum. The museum is well laid out. The building is large and still has graffiti from when immigrants were coming into this country. The displays give you a good feel for how it must have been to be going through the rooms. The exhibits are interesting and chances are you will leave here wanting to find out more about your heritage.
One thing to remember is Security is tight on the Island. Before you board the ferry you will go through security measures more stringent than what you experienced at the airport: x-ray machines, metal detectors, removing belts and shoes. The process and lines are long. Be prepared. Give yourself ample time. The Islands open at 9:30am and close at 5pm.