With regard to quarterly estimated federal taxes, you want to pay at least enough to avoid an underpayment penalty. Generally, most taxpayers will avoid this penalty if they owe less than $1,000 in tax after subtracting their withholdings and credits, or if they pay at least 90% of the tax for the current year, or 100% of the tax shown on their return for the prior year, whichever is smaller.
A very general rule of thumb is to pay 20-25% of your gross income.
If you will still have a state income tax obligation as explained by @Opus 17, don't forget to make state estimated payments too.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
