Must I pay a credit card balance after receipt of the statement, or can I pay as I go?

Yes, you can pay off your credit card as early as you like, even before the closing date. If you do this, you'll see this payment on your next statement, and the amount due will be adjusted accordingly. And as long as you pay the entire statement balance by the statement due date, you won't be charged any interest.

I don't know anything about your particular bank, but in general, yes, you can use online bill pay services to pay your credit card. Your credit card company probably also has a mechanism on their website for paying your bill electronically.

I've used a computer bill-paying service for 20 years, and my Uncle started using one in the early 1980s via a 300 baud modem (I use the same company that he did before he died). It even auto-mails checks to people/companies that don't do EDI. It's great as a centralized payment center, and is agnostic to where I keep my bank accounts, letting me add as many as I want.

Regarding early payments: pay as often as you want; the bank doesn't really care. In fact, for reasons of fiscal stability I pay down our CC every Sunday night. (We'd gotten in deep CC debt, and now I watch our bank and CC balances like a hawk.)

Since this is your first credit card there's a very good reason to not pay before they send your statement: You want to have a balance on the statement date so it gets reported to the credit bureaus. You want that paid-as-agreed to show up on your credit report and if your balance is zero on the closing date you might just get a zero instead. Zeros don't hurt you but they don't help build a credit rating, either.