Don't be afraid to start leading right away if you feel comfortable.

Whether leading or top roping, get lots of milage on stuff that is hard enough for you but that you won't usually fall off of. Falling off the same moves over and over doesn't help you improve as fast as successfully climbing many different routes, in my experience.

When I started climbing, I only climbed sport routes for many years. On those, we would certainly climb until we fell, especially if it was overhung. You have to really be an attentive belayer on sport routes, as falling is common and expected. Toproping laps is a really great way to get strong, too. I tend to do that more than bouldering, actually.