When you get pregnant, is it possible that your skin will break out/hair get more oily?

Hormone changes. Unfortunately, yes due to hormone changes.
Yes. Pregnancy can cause changes in oil production and can affect hair growth and texture. Many women actually experience increased hair body during pregnancy. A good skin/hair care regimen can help minimize any problems. Importantly, many prescription skin medications are unsafe in pregnancy. Review all medications and supplements with your doctor if you are pregnant or considering pregnancy.