Brand new to exercise, just want to share my tiny victory so far and need some advice on continuing

Male that just turned 40, 5'10" 220 pounds, and for the first time in my life I've decided to exercise. Not doing anything crazy, just thought I would start doing pushups. I've never been very motivated or even attempted to exercise, so thought it would be best to start simple. Figured if I'm a little fat I could at least have stronger arms haha, and would see if any other motivation would kick in from there. Well, I quickly learned that at some point in my life I lost the ability to do pushups. I could barely squeeze out even a couple. So for the last week I've been doing "girl pushups," (sorry no offense intended, I don't know what else to call them. I'll refer to them as "half pushups" for the rest of this post). First few days I was barely able to do 2 sets of twenty "half pushups," and that turned my arms to jelly and I was sore the next day. Then it turned to 50, then 60 with much less soreness. I am pretty amazed how fast I improved on a day-to-day basis and could feel results. Today I have already done 60 half pushups and even managed to do 10 regular pushups. It feels really good and I will keep it up for sure.

Is this a good tactic? To keep splitting up the half pushups and and regular pushups? I feel like I need to get my arms stronger in order to complete more regular pushups and get the full body effect of those. Any other simple exercises I can do at home? Not ready to head out into public or a gym or anything. I tried doing situps, and unfortunately I can't do any. I've been having some lower back pain that I need to see a doctor about, and I'm not strong enough to do them anyway. Even trying to cheat a situp by swinging my arms for momentum hurts my back.