I have to admit that even though I lived in Manhattan for a while, I never fully got a grip on where Tribeca was (!).
The times I visited I was with people who knew what they were doing, or I was visiting a particular venue. The Odeon, for example, a classic bistro on West Broadway. Or Walkers, another bar/restaurant with old style charm.
I tried going there recently but wound up near the World Trade Center, walking aimlessly. How embarrassing!
Now, after finding it again, I think the best options are these - go to City Hall and walk west, or go to the Chambers Street subway station. The neighborhood is rather small, so it's easy to miss.
Related posts: Decisions, Decisions, On a Stoop, in Soho and The Best of Times, the Worst of Times.