There are three basic categories to think about: federal income tax, federal self-employment tax, and state/local tax.
The most confusing one for contractors is the self-employment tax. Self-employment tax is FICA (Social Security and Medicare) tax for self employed people. As a regular employee, you would see money taken out of your paycheck for Social Security and Medicare. Your employer would pay additional money for your Social Security and Medicare. Since self-employed people are their own employer, the self-employment tax is you paying both sides of this tax. Last I looked, it was about 14%.
Your federal tax bill ends up being your federal income tax PLUS your self employment/FICA tax. If you expect to owe more than about $1k by the end of the year, then you have to submit quarterly estimated tax payments as someone else mentioned. These payments are balanced against your taxes owed at the end of the year, just like you would normally do with your employer withholdings.
Look up IRS Schedule C for deducting business expenses from self-employment income, and look up Schedule SE for the self-employment tax. These are what you will file with your Form 1040 as a person with self-employment income.
It's a 3 month internship at this point so I'm really not certain as to how to go about this the way you've laid it out here.
That's the thing, there's no particular one way to go about it. If you're employed the rest of the year, you can probably just bump up your employer withholdings during that time and not worry about the self-employed months until you file your taxes next year. If you don't know what the rest of the year holds, then you'll want to pay as you go to avoid penalties.
If you're not sure, I'd recommend saving about 1/4 of your salary to cover tax payments and filling out the estimated tax form for whichever quarters apply. Please note that that's a ROUGH estimate and I'm not an accountant.
Congrats on the internship BTW, I forgot to say that before.
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