We always get off in Nassau, and have several spots to visit. First and foremost, a stamp in our passports (which can be done right outside Festival Marketplace, where you exit once leaving the pier). We also like strolling over to Fort Fincastle (gorgeous views from up there) and Queen's Staircase (very beautiful spot). If you don't mind walking, it is about a mile walk there and a mile back. Very nice stroll, especially on a nice day. There is also a public beach (Junkanoo Beach) that is about a 12-15 minute walk from the pier (or you can take the jitney, I believe #10 there for 1.25/person). And of course, no trip to Nassau is complete without a visit to Tortuga Rum Factory for rum cakes! This is literally a couple of blocks from the pier. Oh, and if you are interested, Hard Rock is right across the street from the pier, as well as Senor Frogs.
Let me know if you would like directions to any of these places; ask away and I will provide!