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?
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.