The Statue of Liberty is one of the most recognizable World Heritage Sites in the world, and the only World Heritage Site I have ever seen someone emulate in costume (whether in Times Square or on US street corners in spring hawking tax preparation services). I've often seen the statue in New York harbor passing by or through New York City, but the only time I visited the statue was on Leap Day 2008. It's hard to appreciate the size of the statue until actually visiting Liberty Island and walking around the base. I didn't have tickets to climb the statue, but I did get a ticket for the pedestal and the museum inside, which I enjoyed (the old museum has since been replaced by a new one that opened in May this year). The statue itself was worth visiting for what it has meant to people throughout its history, and how people still view it, to include:
1) A symbol of liberty, of freedom for all men -- an unalienable right America's founders recognized in the Declaration of Independence, even as the country still strives to reach the ideal that "all men are created equal".
2) A symbol of immigration and the American dream, as the statue was part of the first view of the United States for thousands of immigrants arriving by boat, seeking a better life for them and their children. This hope has been immortalized in Emma Lazarus' poem The New Colossus, which Kyle has included in his review.
3) A symbol of the friendship between the US and France, America's first ally in its fight for independence from colonialism.
As Americans have debated what type of nation they are over the past several years, I would hope the Statue of Liberty reminds the US that it a nation built on hope, not fear.

Logistics: The Statue of Liberty can only be reached by boat; tickets for boat access to Liberty Island and for access to the pedestal and crown can be purchased online or at the official vendors in Battery Park in New York or Liberty State Park in New Jersey (tickets should be considered in advance, particularly for the crown). Like others, I highly recommend a visit to nearby Ellis Island, with its excellent exhibitions on immigration.