In NYC, the easiest way in is through Local 1's apprenticeship program, but, alas, you just missed the once every three years test they gave back in August. They give this long test of shop math, spatial and mechanical reasoning, tool recognition,
etc., and then the top 40 scorers get added to the apprenticeship list.
As openings come up on the list, those people get called, and then pay initiation and apprentice dues, and work in an apprenticeship for 18 months to three years and, after successfully completing that, are granted full membership.
I'm actually on the apprenticeship list, myself, although in the meantime I'm a full member of Local 260 out of Lake Charles, LA for the show I'm touring with right now.
There are other ways in, but from guys I've talked to, that's the easiest way.