I am getting fatter and would like to fix that
Parroting everyone else here- start addressing this first with diet if you are sedentary. Your joints will thank you.
If I was to only do sit ups, pushups, squats, planks and wall sits, are there any key muscles that I would be neglecting?
Chiefly your back, glutes, and forearms. Depending on your form, there are tons of accessory muscles that get lost in the sauce.
What bodyweight exercises would you recommend to target those areas?
Stretching, lunges, burpees, and pull ups.
Fat == sedentary == not limber
Not limber == some variations of exercises will feel prohibitive because you feel like you can't do them.
Eg- you'll want to keel forward while squatting. This splits load between your lumbar and quads instead of glutes and quads.
Chest high and back straight during a squat will move the load from your lumbar to your glutes.
Lunge variations are free and they make all of the difference in terms of muscle groups targeted, but there are brain games here. Try doing 5 lunges by stepping forward. Now contrast that with how it feels to do 5 by stepping backward. Contrast that with a split lunge/Bulgarian split squat.
Burpees are great all the way around. You'll look and feel retarded, but they'll work just about everything and get your HR higher faster than just about anything without equipment short of sprinting.
Pull ups are a slog if you're fat and they don't fit your question, but they're worth bending your rules once you're at a good weight to do them. If you decide to move to doing bar work, you will appreciate having grip strength.
How should I plan out my sets of these exercises? Should I do the same routine every day?
There are a few schools of thought on rep and set schemes. None are magical. 7 reps won't turn on your balls to make more testosterone. Just go until you feel like you've got two left, stop, and then do that again until you can't.
If you don't want something like that, you can do something like:
Pick a period of work (30 seconds or something)
Do the exercise and do not stop for that period
Rest for half of that
Repeat
There's supposedly science behind the work, rest for half the work period, and repeat thing. It's called Tabata and usually goes (20/10)×8, but I'm not going to do a Tabata sperg.
the most important thing is that you find something you enjoy doing
This is important.
Can't go wrong so long as you don't hurt yourself.