Push-ups ARE weight training. Actually, the proper terminology is “resistance training” and perhaps that’s why you’re confused.
What is Resistance?
“Resistance” refers to any force your muscles attempt to overcome in order to move. Dumbbells, machines, exercise rubber bands, your body, and gravity are all forms of resistance. When you do a push-up, you press your hands into the floor and straighten your arms to lift your body weight (AKA resistance) upward.
All forms of resistance training involve performing a few repetitions (usually between 6-15) using relatively heavy resistance, as opposed to aerobic activities like biking which involve high repetitions and low resistance. You should use enough resistance to make each set challenging enough to tire you out and, in effect, stress your muscles. This will literally tear your muscles apart.
You need to give the muscles you work at least 48 hours of rest between resistance training sessions. It takes that amount of time for them to repair, rebuild, and recover. By the way, the same holds true for a particularly intense aerobic session. Most aerobic sessions aren’t intense enough to tear up your muscles (at the microscopic level). But notice that when you do all-out wind sprints or heavy duty hills, your muscles are sore for a couple of days afterward.
Eventually, your muscles adapt to the resistance they push against by becoming stronger. That’s why you can lift progressively heavier weights. This is also why, eventually, you’re going to have to look for a way to make those push-ups more challenging.
Increasing the number of sets you do, slowing down your reps or adding an exercise band is three ways of making push ups harder.