I'm in California and I just received a 1099 misc from my employer.

Do I have to pay CA state taxes on top of the 15.3%?

How/where do I pay the 15.3%?

Do I just send a check to the IRS/State?

You need to file Form 1040 with Schedule C and SE to calculate your tax liability.

If you owe money after the calculations are done, then mail a check made to "U.S. Treasury" with the tax return.