First, you'll need to understand how a credit score is determined. Once you know what makes up your credit score, you can make decisions to help your credit score.
As for paying off a balance in full, it is a wise option to not only help your credit score but you'll also save money by avoiding interest on the balance. Using your credit card for small purchases each month and paying off the balance can also be a good way to increase your credit score since it will show that you are always ontime with your payments.
Credit card companies report payment and balance information monthly to credit agencies, so you should see the change (hopefully an increase if you've done all the right things) to your score within a few months. For more help, read about simple ways to increase your credit score.