Things are generally fine. A credit balance is not a horrible thing.
The argument against maintaining a credit balance is that you are essentially loaning the credit card issuer money at 0% interest. You probably have alternative investments that would pay better interest, so it's usually better to park your money there. All that said, it's unlikely that the interest on whatever balance you have is enough to be more than pennies.
The way that a credit card works, you run up a balance in one period. Then there is a grace period. If you don't pay off the balance during the grace period, they start charging you interest. You also may have a minimum payment to make. If you don't make that payment, they'll charge you a late fee.
The typical period to rack up charges is from the first to the last day of a month. The typical grace period is through the 20th or 25th of the next month. Your card may be different. So check the documentation (user agreement) for your card if you want the real data.
It sounds like you paid off some purchases while you were still in the period where you rack up charges. While those purchases were posted to the account, they may not be counted in the balance calculation. If your credit balance exactly matches the payment you made, that's probably what happened. It's also possible that you overpaid the balance. If your credit balance is just a small amount, that's probably what happened.
If you really want to be sure, you should call the credit card issuer and ask them. At best we can tell you how it normally works. Since this is your first month, you could just wait for your first bill and respond to that. So long as you pay off the entire balance shown there by the deadline, everything should be fine.
Don't wait until the last day to pay. It's usually best to pay a week or so early so as to leave time for the mail to deliver the check and for them to process it. You can wait longer for an online payment, but a few business days early to give you a chance to handle potential problems is still good.