extremely weak upper body

Hi all,

My upper body is super weak and I can't support my own body weight. I don't fall over while doing burpees or mountain climbers exactly, but I have never been able to do a full push-up and I can barely do 2 or 3 knee push-ups. Wondering about the best way to strengthen my poor arms. I've tried bodyweight, but it's not super effective because I can't do half of the exercises. Strength training provides more of the "burn" I'm looking for, but I'm worried that it will give me "artificial" strength because I don't want to be able to lift weights but not my own body weight. I lift super light though; either 8 or 16 pounds total depending on the exercise. My goal is to be able to do 20 push-ups without falling over. Suggestions? Should I do bodyweight first and then move on to weights? Thanks