I did a 10-week internship in Switzerland and earned 5770 CHF. Do I claim that on my taxes? Does Switzerland already figure taxes and therefore I do not have to pay because I can not be taxed twice? I live in the US.
Were Swiss income taxes withheld from your wages (i.e. the 5570 CHF was what you got in hand after taxes were withheld), or was that the entire amount that you earned -- no deduction for any kind of taxes? Regardless, as a (presumably) tax resident of the US (or a US citizen), your Swiss wages need to be reported to the IRS and you have to pay US income tax on those earnings. You can get a credit for taxes paid or withheld in Switzerland against the US income tax due on those Swiss wages if you file the appropriate forms, so that you avoid double taxation of those wages.
In general, yes
The U.S. taxes income of all citizens, regardless of where the money was earned.
I can not be taxed twice?
This is patently false. You can be taxed twice in the U.S. alone via state income tax. In almost all cases, you can write-down your income based on other taxes paid.
You can attempt to do this alone, but every ex-pat I know pays an accountant to do their taxes. It's not a common case, and you certainly don't want to screw up as your return will almost certainly be scrutinized.
As of about a decade ago (when I last worked there - I don't know if laws have changed), you will have to start by figuring Swiss income tax - both federal and for the canton you worked in. (Swiss tax returns were dead easy compared to US ones.) This might have been deducted from your pay.) Then you can take a credit on your US tax return. If it's a small amount (likely in your case), it can be taken on Form 1040 Schedule 3, otherwise you have to file Form 1116: https://www.irs.gov/individuals/international-taxpayers/foreign-tax-credit
The upside of this is that you may be able to deduct a standard per-diem amount for living expenses, which is likely to be a good bit more than you actually spent as a student.