Exactly. Take the subway to a part of the city you don't know, and just stroll. Stroll through union square, the village, soho - you will always find something interesting, cheap, and fun to do. Even if you end up doing literally nothing, it's always fun.
Stroll down St. Mark's (at night, of course), go to the Life Cafe (if you're a renthead), just do whatever.
Get lost in Central Park. Go see Regis and Kelly.
Oh, and as far as the museums go - the mets are donation only. I once had to go for school and had literally $1 dollar to my name. If you're really short on cash, you can give them $1. You might get a glare, but you'll get over it.
"If you are going to do something, do it well. And leave something witchy."
-Charlie Manson