Anthology Film Archives is a great place to check out for a variety of independent films. I've only taken advantage of the movie theater there, but there's also an art gallery, a film library and, you guessed it, a film archive.
Best yet, tickets are only $8 ($5 for members, and cheaper for everyone for some films).
Anthology Film Archives hosts a lot of festivals, and they run the gamut from Chinese Horror to documentaries about war by people who have lived through it to American Independents - I've seen an unfinished film by a Bosnian director about going back to reclaim her family's home, and a short entitled "12 Beers in 12 Minutes" (which was just that and ended up getting very messy in the end.)
There are fliers and schedules in the lobby as well as information on the website. The only thing is that you have to be a little flexible in terms of scheduling, things don't always run smoothly and I've had show times pushed back by as much as 1/2 an hour.
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