How soon will my credit score improve?

My credit is horrible - it's between 550 - 600. I have about $2,000 in debt from credit cards I got when I turned 18. I'm 22 now, have a secure career and am about to pay all of my debt off in full. I am also going to get a secured credit card through my bank (Wells Fargo). I know to charge and pay off balances in full monthly, but how long will it take to bring my score up? I'm planning on buying a house in about 2.5 years. Assuming that I don't incur any more debt and am responsible with maintaining my cards, will my credit be at a good point? Would it be bad if I bought a car before buying a house?