Do not use KIlz would be the first step
Use drop cloths to cover stuff up and get some Gardz for the primer.

Wear a painters cap or a kerchief.

Gardz is very thin and watery. Roll slowly and cover for any splashing.

I wonder if the woven rollers splash less.

Use a micro fiber cover

Use a primer and paint that doesn't spatter. It isn't the roller, it's the product you're applying.
You might also try a pad, but only for the primer.
No to Kilz. Use Zinsser products. You'll like the results.

1 - Don't spin the roller! Stop the roller on the wall or ceiling and lift it off instead of using that "kid with a toy car spinning the wheels "motion at the end of a stroke.

2 - Keep the roller wet - a common painting novice mistake is to keep painting when the roller has too little paint on it (that "frying bacon sound" as you roll means it's dry and it's flinging droplets) and pushing harder and harder to get the last bit of paint until the roller is totally dry.

One roller-ful will do a 2ft x 2 ft up to 3ft x 3 ft area. Reload the roller after it, even if you could squeeze out a couple more square feet of coverage.

3 - Wear a hat, shower cap or something on your head.

4- Use dropcloths.