IV's determine the Pokemon's characteristic, and EV's don't affect IV's. You'll have to stick with breeding

But how does it all work with breeding? Does it carry over? or is it completely random?

3 IV's are randomly selected from either parent to the child, and power items pass certain ones on.

Doing that, I was able to hatch a lot of eggs to get a Shroomish with 31 in HP, Defence, and Speed. However, the other 3 are up to chance to hatch a lot of eggs and look for the 3 IV's you wanna pass on (Generally HP, Atk/SpAtk, and Speed). Then hope for the rest, but usually the opposite attack stat won't matter so it's not too rare to get a near-perfect pokemon in 100-200 eggs. Pair that with Everstone for a 50% chance of getting the nature you want for as close to perfect as you can get without RNG Abuse

It's as stated before, completely random, the only way to get exactly what you want (without Action Replay) is to

find two suitable parents with high Iv's

get atleast 50+ eggs

hatch them and hope you get lucky

TO THE DAY CARE CENTER

Random.