Advice for ultrabeginner?

I am female, 22, 5'7", and around 115-120 lbs. As long as I can remember, I have been unable to do even a single pushup. I'm fairly active (probably not at all compared to most of you, though) and even do some rock climbing, but it seems that some crucial muscles are pathetically weak. I'm wondering whether anyone here has any advice, or ideas about how to work up to it. I do not have a gym membership, but my roommate has some weights he'd probably let me use if that would be helpful. I'm pretty terrible about sticking to exercise plans, so I'm trying to set a very clear and attainable goal (i.e., be able to do a push up).

Thanks so much!

UPDATE: 2 days later, I can do a (still pretty shaky) push up. Awesome. Thanks for all the help!!