Raising my credit score!?

I tried to get qualified for a home loan with a co-signer, but couldn't because I had a credit score of 598 instead of a 600. We just bought a car and are getting ready to make our first payment. My question is after I make a couple car payments, and pay off my most recent debt in collections, would that raise my score? And if so how long would that take? I only need to raise it a minimum of 2 points.