I use to only do calisthenics, and running for my fitness health. Usually running 2 5ks, and several single mile runs ending around 10-12 miles a week.
I did not have any help doing so, and after stopping for stationary cardio options I want to get back into running. I would like to be able to first get to the point that I can do a 5k without stopping, and then get to a sub 22 minute 5k.
Any tips to progress into it without burn out?
There really is only one way to get better at running, and that is run more. However, people often run their hard runs too easily, and their easy runs too hard. A favorite coach I know would say "Run. Run lots. Mostly slow, sometimes fast."
For your initial goal, I would do a proven program such as couch to 5k (Often abbreviated C25k), which gives a good 10-16 week progression of run/walk to the point where you can run a 5k without stopping.
After that, I often recommend a 3-2-1 program. If your long run is 5k and that takes you 30 minutes, then you would have 1 run of 30 minutes a week, 2 runs of 15 minutes, and 3 runs of 7.5 minutes. Increase no more than 10% total time a week, and every few weeks don't increase for a week or two.
When you get to the point where you are reliably running 25-30 miles a week, then you can start adding speedwork towards your 22 minute goal. There are two types of speedwork, interval and threshold. Threshold is slightly less than your race pace, done for long periods and shorter recovery. This allows you to run longer at a higher pace. Intervals are shorter distances, and done at maximum speed (Sprint rather than race pace), with a longer recovery to enable you to do them at max effort.
If you haven't really run before, I would first recommend the C25k or similar, build up to 25-30 miles a week, and then look at an online training plan to get you to your goal time. It's hard to know how to program your own workouts without a lot of experience.
