How do I know how far away from the wall to make the hole for the drain so I don't end up with a huge gap behind the toilet?

Since your post said "moving" toilet, you have it? Yes or no. Toilets can be 10 or 12" rough in, most commonly 12. Measure it. If not, get a 12" rough in and make the floor flange centered at 12" from the wall stud behind it. You can also set the bowl on the floor, temporarily set the tank on it, place it where you want it and set the floor flange centered on the bolt holes. Leave 1/2 to 1" behind the tank to the wall

how do I know how far away from the wall to make the hole for the drain
For a 12" bowl make your hole 11 1/2" to center off the finished, (drywall) to the center of the hole. Good luck, Tom