I know I can plug in a 3.5 mm headphone plug into the Switch, but is there a way to sync Bluetooth headphones instead?
According to the Wikipedia for the Switch:
The unit does not directly support Bluetooth wireless headphones, though they can be connected with third-party wireless adapters via the headphone jack.
Without the use of a third party connectivity device that you plug into the headphone jack, the Switch does not natively support Bluetooth headphones. One article I found shows how you can use one of these devices. It should be fairly straightforward. All you should have to do is pair the Bluetooth device with your headphones.
Unfortunately, this means you will have to make an additional purchase. Also be warned that each Bluetooth device will be different in how you pair the device with your headphones, and the article says that the one they used had a decent amount of audio delay as well.
USB audio is now supported in both docked and undocked modes, meaning various headsets which use a USB dongle for connection - such as the Playstation Wireless Headset - are now supported natively on the Switch. You just plug it in and - hey! - it works. [Nintendolife]
This means you need a headset that ships with its own USB dongle. Buying a regular Bluetooth dongle won't work, as there is no way to pair the headset to it. Also note that most if not all such dongles use USB A instead of USB C, which means you can plug them into the dock, but you will need a USB A to C adapter for the Switch itself.