Oils aren't the culprit - well some types are but not all....
I used to have acne prone skin and now it's combination since i've gotten it under control.
I start with a Pre-Wash Oil, then I use a natural soap (either Green Clay Soap, Apricot Scrub, Oatmeal, or Tea Tree Oil depending on how my skin is that day), followed by a really light 100% natural moisturizer. All the products are handcrafted by Athena's Olive Tree (
Athena's Olive Tree - Providers of Bath and Body Products and Gifts) and they all work really well. I think it's because of the oil and moisturizer plus the fact that everything is natural.
Find something that works for you or contact a professional and ask for their advice. Athena's Olive Tree will customize products for your skin type if you contact them but I didn't need anything custom. Always use a moisturizer (even if you have oily skin), always drink a lot of water, and never never skip washing your face in the morning or at night. If it gets really bad then start changing your pillowcase every other day too.