I agree completely. TKTS only offers shows for which there are a lot of unsold tickets, i.e. usually not good shows. You'll have much better luck going to the box office shortly after they open in the morning -- even sold-out shows have cancellations, and all theaters release "deadwood," tickets held for the producers, reviewers, etc.

DON'T go to the Statue of Liberty. There's little to see there and you'll get a much better view of the Statue from the Statan Island Ferry (which is free). DO go to Ellis Island, which has an interesting museum about immigration to the US and to NY.

If you have any relatives who immigrated to the US through NY, the Tenament Museum is fascinating.

Check out the Museum of Modern Art, which just reopened.

As someone else suggested, the Metropolitan Museum of Art is a must-see if you have any interest at all in art.

If you like opera, a pilgrimage . . . um . . . visit to the Metropolitan Opera is a must. There are frequently free concerts at Lincoln Center where the Met is located.

I'm a former New Yorker and still go back a couple of times a year, so, unfortunatley, I can't suggest many of the more touristy kinds of things.