I am not intending to be rude at all, just give you honest advice. A lot of people think the cinema business is a lot more flexible than it is. It really isn't. I think Bob's advice to you is good, and you should listen to that, but even Bob is saying that his theater ran primarily films, with the occasional live production.
Okay, to answer your questions specifically:
What equipment do we need:
If you want to run new (or newish) films you need to install a digital system. It will cost you no less than $60k, probably much more because I'm sure your auditorium is fairly large. Showing newer films (not on DVD yet) will provide a fairly stable basis for your business, but it requires some commitment and more rigid scheduling.
If you want to show DVDs occasionally, you'll need to invest 5-10k on a decent media projector. This projector will not be able to play new films. The link Bob gave earlier is a good place to start. You'll still have to pay a rental on them, and from my experience people aren't that interested in coming to the theater to see something they can rent on Netflix, or that doesn't have mass appeal. We show art films on DVD once per month. We have a very loyal and enthusiastic following of about 50 people. You could totally make this work, but it is kind of a hard sell.
where do we go to rent movies to play?
We use a booker, which is right for us. There are a lot of people on here who book films themselves. However, as a non-profit, I think you can rent older films through Swank or Criterion. If you google them, you'll find contact information. Either way, even on DVD's you have to pay a rental and the
minimum you can expect to pay on any film rental for a theatrical presentation would be $250.
I truly wish you good luck. I love historic theaters, and it is awesome that your group has rallied to keep your theater open. I hope that your group will strive to make good
business choices so that you can afford to keep the facility maintained and the doors open.