What's the deal with sprinting? How does it work and when should I use it? Same question for sidestepping. I don't see any reason why I'd try to dodge instead of shoot, unless I have something that lets me reload while I move.
That's the MAIN reason, but it also really helps you stay alive if you run away. Which in harder situations you'll be doing a lot if you play like me. Sprinting also lets you kite enemies by being at the edge of their LoS. So you see an enemy, you sprint out of LoS, shoot where you think it is, it moves into LoS, you back up, etc.
The basic strategy if you want to use a shotgun early (not even with skills mind you, shotguns can be useful without them early on), is if you get caught in the open you need to run back to any corner, and then start blind firing shotgun shots around the corner. Since shotguns never miss, the out of sight accuracy penalty is meaningless and the shotgun's knockback will make enemies take 2-4x as many moves to reach you. So you need to be able to get back to the corner, ergo sprinting and dodging.
Sprinting also lets you kite demons, which is hugely important earlygame. This is sort of an advanced strategy (by which I mean you don't need it except on the two highest difficulties) but to be faster than demons with no speed boosts you need to sprint
and take off your armor, which is important to know because the game will sometimes spawn "vaults" that contain 6+ demons which will rush you after you open the door.
Dodgemaster (which makes your first sidestep per enemy per turn 100%) and Fireangel (which makes you immune to splash(!)) totally changes sidestepping. Now against certain enemy types (the "demon noble" archetype in particular) you can move just to re-position yourself with no concern about damage. So its perfectly valid to just zigzag towards them until you're in point-blank and then beat them to death or blow them away. You can also cause enemies to hit each other.