As for me, I can say that I have been to Staten Island about a handful of times. I think that's in part because the commute between Staten Island and the rest of the city can be a bit of a trek, even with our great public transportation system.
However, I will say that Staten Islanders do have one great transportation resource that I think benefits both locals and visitors to NYC alike: the Staten Island Ferry. It takes people from Lower Manhattan to the northern tip of Staten Island for FREE. If something is free in NYC, I advise you take advantage of it. :) As practically a life-long New Yorker, I'd never taken the ferry and decided to make the trip during one of our heat waves this past summer.
On the Manhattan side, there is a brand new terminal that opened in 2010.
After entering the terminal, you'll go upstairs and see a waiting area with electronic signs showing the next ferry's departure. People tend to start lining up about 10-15 minutes before the doors open allowing you to board. With it being free, you'll see both visitors and commuters, so be aware that it will be fairly crowded even in the middle of the day and/or in the middle of the week.
The crowds can get pretty thick.
Boarding...
The ferry leaves its Manhattan terminal
As a visitor what you're really taking the ferry for are the views of Lower Manhattan and everything in between.
Battery Park (I got married here :)) and lower Manhattan
The Brooklyn Bridge and Manhattan Bridge to the north
The New York City Water Taxi (another great way to see NYC from the water)
You might recognize this as you make your way to Staten Island....
Happy Belated 125th Birthday to Lady Liberty (it was last week)!
With so much to see, you actually can forget that you are crossing a working harbor.
Soon Manhattan starts to fade away, and Staten Island comes into view.
Another ferry waiting on the Staten Island side. I like that the ferries are painted a happy orange color.
I guess you might be thinking that you should just turn around and go right back to Manhattan, but I would urge you to take a few minutes to walk around the area right outside of the ferry terminal on the Staten Island side, which lets you off in the St. George neighborhood.
The terminal does provides transfers for the Staten Island Railway and buses that travel south into the rest of the borough, but you can still see some nice views here as you walk on the esplanade. I will warn you that it is a little industrial though (but, hey, you're in New York City).
As you continue to walk on the esplanade, you'll come to the Staten Island 9/11 Memorial. I have to admit that I was not aware that this existed, but I am glad that I found it.
Once you enter between the pillars, you will see the names of those who died on 9/11/01 who were Staten Island residents. I think the memorial is moving in its simplicity.
Practically across the street from the memorial and the ferry terminal is the baseball stadium for the Staten Island Yankees, a minor league team that feeds into the New York Yankees and one of two minor league teams that play in the city (the other is the Brooklyn Cyclones based at Coney Island in Brooklyn).
That's really it for things to see that are a short walking distance from the terminal, but even for someone like me who's never been on the ferry, I thoroughly enjoyed my ride. If you're ever in New York City and looking for a quick, pleasant excursion through New York Harbor, hop on over to Staten Island.