The first part is getting routine, the second is getting the form which is where I think I am at right now.
I commend you for starting and sticking to an exercise routine. You are absolutely correct that starting/sticking to a routine is both difficult and important.
But I have some unsolicited advice for you, based on 20+ years of bodyweight exercise experience. Feel free to ignore any or all of it. I'm only telling you this because I wish someone would have told me some of this stuff this long ago.
1)
I would emphasize form over almost every other aspect. When I was much younger I was all gung-ho about increasing my pull-up and push-up reps, and I didn't think about form at all. For the past 10+ years I've dealt with shoulder pain and mobility issues as a result.
Now that I'm older and wiser, I'm meticulous about form. I perform my reps slowly and deliberately, and stop as soon as I feel I can't maintain excellent form. My mind is always focused on the tension of the muscles that are supporting the movements, and that helps keep the form where it should be to prevent injury.
2)
I would not attempt high/fast reps until I had perfected my form. It's way too easy to develop bad form habits when you're focused on high reps. High reps implies faster rep rate, and it's very difficult to keep tabs on your form when you're moving quickly. Once you've mastered your form it's a bit easier to move quickly and keep good form, but when fatigue sets in at high reps you're still susceptible to bad form creeping in.
3)
I would not work on horizontal pushing without also working on horizontal pulling. You're going to end up with muscular imbalance in your upper body, and that is going to cause progressively worse posture issues over time. If you're going to be spending a lot of time on push-ups, you should put forth a similar effort doing inverted rows. While you're at it, for overall upper body muscular balance I would also recommend vertical pulling (pull-ups/chin-ups) and pushing (overhead press/handstand push-ups). Just get a dang pull-up bar already -- it's the only piece of exercise equipment you'll ever need. :D
4)
Don't forget to train your lower body as well! Squats, lunges, and running are great choices.
5)
YMMV, but personally I see relatively little value in high/fast reps for bodyweight exercises. You will increase your muscular endurance for those specific exercises, but that will not be particularly useful in most people's day-to-day lives. What is (generally) more useful is increased strength, which allows you to move yourself and things in your environment with greater confidence and with less risk of injury. Secondarily, maintaining (or increasing) your muscle mass is important to your long-term health, and once you hit 40-ish, it will require more and more effort to prevent losing muscle mass.
None of this is to say that muscular or cardio-vascular endurance isn't important as well. It definitely is. But -- Herschel Walker and Jack Lalanne notwithstanding -- I think you're leaving a huge portion of the potential benefit of exercise on the table if you focus exclusively on reps.
6)
Don't do sit-ups. Especially if you've already had back problems. You can get a much better ab workout by doing hanging leg raises on the pull-up bar.
7)
Rest. It is possible to exercise every day, but your body really does need time to rest and repair the muscles. Not only that, but your nervous system needs a break from time to time. I'd recommend not hitting the same muscle group hard on back-to-back days at least. And I think it's beneficial to just give yourself a complete break from strenuous exercise at least one day a week. You can get away with all kinds of crazy stuff when you're younger, but as you get older, the periods of rest become even more important.