When self-employed you are required to make quarterly estimated tax payments. While there are "tools" out there to help one figure out how much they need to send the IRS each quarter, it's a waste of your time to use them. The general rule is, if you send the IRS 20% of your *gross* earnings each quarter, then come tax filing time you'll be fine. If your state taxes personal income that same general rule holds true for state quarterly taxes also.
For your vehicle mileage, be aware that commuting miles between your place of residence and your primary place of employment are never deductible.
