Headset
 

If I get a new headset with a mic, should it work on Linux Mint?

Yes, It should.

Provided that:
A: the headset isn't USB -- this may require special drivers.
B: it has appropriate Mic and Headphone connectors
C: Your sound card is working in Mint.

What type of headset? USB? A standard headphone/microphone combination? Bluetooth?

All of those would be supported, but it would help if you clarified what exactly you are looking to get so you could be further assisted with the setup of the device.