If you're playing games where directional audio matters, I would strongly recommend monitor headphones. It makes it much easier to figure out exactly where sound is coming from. I have Sony MDRs and I love them.

For a mic, there are some great options out there that are reasonably inexpensive. I have a Blue Snowball that's lasted the better part of a decade and still works great, and I know that there are others at a similar price point that might even be better by now. Check out reviews from humans, whether text or video. Boom arms go a long way toward minimizing background noise from desk vibration, and also make it easier to adjust the placement of the mic.