Main thing you need to realise when wanting to take part in astrophotography is that the Earth moves. So you can do long exposures but it's going to result in light trails.
So first thing is first. The wider your lens, the longer you can keep your shutter open without any noticeable light trails. 30mm (FF Equivalent) will allow for 15-20 seconds, 50mm (FF Equivalent) you could get 10 seconds out of maybe. Ideally though if you have access to a 14-15mm lens then you can get exposures as long as 40 seconds (so with a camera like the T3i you're looking at 10mm/18mm/30mm).
As for the rest of your settings, ISO 200-400, Aperture (if you're pointing straight up into the sky) f/5.6-f/8.0
Also, manually set your focus distance to infinity (or very slightly less depending on the lens), and make sure to turn off autofocus.
As others have said, 5 to 20 seconds is probably the range of exposure time that you want to work with. Play around with setting aperture and ISO around that, and see what you like best. Try playing around with post-processing on your shots to see if you can work with the noise at high ISOs, but it might be hard since you're going to be throwing away detail of the celestial imagery itself.
@smitchell22: The difference in brightness between the stars and the moon is too great for the dynamic range of a camera to handle, so you can only have one or the other exposed properly in a single shot. The only way around that would be multiple exposures with the same framing, one to expose the moon properly, one for the stars, and then combine them with an editing tool. You need to get the shots in quick succession, since both your vantage point (the earth) is moving relative to the stars, and so is the moon.
@CROM: Thank you for all your advice. It's very informative.