We just started, so there is almost no income and many expenses currently on the books.
You need to speak with an accountant. Or at least hire a bookkeeper Unfortunately, it's another expense, but a needed one.
I am not an accountant and this is not "professional advice."
I do have an S-Corp, though.
You need a corporate accountant. A public accountant cannot do corporate taxes.
You pay by the March 15 deadline like everyone else. UNLESS you are earning more than a certain amount a year, which your account can advise you of--then you will likely pay quarterly remittance which is an estimate based on revenue and then adjusted at fiscal year-end. If you're earning under say $200K, don't worry about it: you're still on the standard tax plan.
The return is due March 15 for the previous year. Your S-corp doesn't pay taxes, it files a form 1120S informational return. The 1120S produces a form k-1 that states your share of the companies income and other items. You report that k-1 on Schedule E of your Form 1040 in April. You have to file the 1120S every year whether you made money or not. Definitely find a professional, the 1120S is not easy to do.
I have an S-Corp and several other corps, but I'm not an accountant.
Definitely hire a qualified accountant, as the various state and federal dates and taxes can be confusing. You can also use a Registered Agent service that offer notifications and managed reports so that you don't miss dates. Companies like MyLLC.com (I am not endorsing this - just an example)