A good rule of thumb is to wait until the muscles you have worked are not sore anymore or considerably less sore than they were
- the day *after* the workout - which is usually when DOMS (Delayed Onset Muscle Soreness) begins.
As for reps - it really really depends on
1) Whether or not you are using good form (not cheating using flying elbows, lifting with your back, swinging, using legs to help push up the weight etc)
2) The speed of your reps. For example, if you can do 20 military paced pushups - trying instead taking 5 seconds down and 5 seconds up for each and every rep (for a total of 10 seconds for a single pushup). I assure you you won't be able to do as many going that slow. That applies to all weight bearing exercises.
Working to exhaustion is good if your goal is to gain some strength - but you do need to allow some time for your muscles to repair.