Flying out of the US
When flying out of the US, you'll need to provide a negative COVID-test, if you haven't been fully vaccinated. If you have been fully vaccinated, you don't need a test unless the destination that you're flying to requires it.
Whether you'll need to quarantine or provide a negative COVID test when you arrive at the destination that you're flying to will depend entirely on that destination.
Arriving in the US
If you're not fully vaccinated, when arriving in the US, you'll need self-quarantine for 7 days IF you get a COVID test 3–5 days after travel. If you don't get a COVID test, you'll need to self-quarantine for 10 days.
If you're fully vaccinated, you'll still need to provide proof of a negative test but you won't have to quarantine or self-quarantine upon arrival.
There are a number of countries that are on the US red list for international travel. This means that you cannot fly to the US from these countries. Please check this CDC red list guide for the most up-to-date information on red list countries.