I rarely see anyone on YouTube using a fancy smancy microphone - everyone is using ones that are like, 1000 or less. I got started with a 30 dollar USB microphone before I upgraded to an audio technica, but if you have a bit more of a budget I'd definitely say a Blue Snowball, Yeti, or MXL008 to get you started. I'd say not to a headset for recording a song, because it'd be really rough on your P and B sounds lol. But yeah - like the previous poster said, a Blue Snowball could work if you want a better sound than your headset.
I also know some people who started just by using their rockband USB mics to start covering stuff, so you have a lot of options in a lot of price ranges
I also want to point out - you'll get the best audio quality if you record your audio separately from your video, mix the audio with the backtrack you're using, and then sinc it up with the video. If you want a live coffee shop performance type feel though, then recording yourself singing live would probably be fine