You can walk almost anywhere outdoors on the Yard. You can also visit the public areas of Bancroft Hall: the Rotunda, a mock-Mid's room (near the main office), and Memorial Hall. You can usually visit the USNA chapel, except during services, but they close pretty early (~3 PM, I think). The crypt of John Paul Jones (well worth your time) is in the basement of the Chapel. The USNA museum, which was closed for renovation for more than a year, is now open again. The USNA Visitors Center has interesting exhibits and a nice gift shop where you can stock up on everything USNA (apparel, bumper stickers, books, etc.)
Anyone 16 or older must show a government-issued photo ID to walk on.