So I want to increase the number of pull-ups, push-ups, sit-ups I can do.
I've been leaving an arbitrary 2 minutes in between sets. Is this the ideal amount of time?
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Two minutes is fine. More rest means you'll improve your muscular endurance and recovery time less, however your strength will improve more.
People training for strength should take 3 to 5 minute breaks, or even more in some circumstances. People training for fitness, hypertrophy, or endurance might want to work shorter rests--2 minutes, 1 minute, or even circuit training, where one does another exercise (e.g. jump rope or air squats) instead of resting.