Don't worry about noise cancelling if you aren't in a consistently noisy environment (public transit, airplane, etc). It'll just make the audio quality suffer and a lot of them can't play music when the batteries die. If you don't want to hear outside noise, you just need properly isolating headphones. I recommend open headphones over closed ones for gaming, but it's a matter of preference.
Here's some setups:
- Philips Fidelio X11, Blue Snowball microphone
- AKG K612 Pro, Objective2 amplifier, Blue Snowflake microphone
- Sennheiser HD 558/598, FiiO E7, Antlion ModMic
- Audio Technica AD900X2, Blue Snowflake microphone
- Beyerdynamic DT880 Pro, Objective2 amplifier, Antlion ModMic
- Sony MDR-MA9003, Schiit Magni4, Antlion ModMic
1: Does not need an amp, does not need its cable changed. Don't listen to the people complaining about it.
2: The A900X is closed, if that's more your fancy.
3: Discontinued; price will fluctuate.
4: Might not be necessary for the MA900s, but I included it regardless.
Some of those setups go out of budget, but feel free to mix and match microphones and amps, as long as you have an amp if your headphones need it. The DT880 Pros and K612s need them. The Fidelio X1s do not, and I have amped them from a Little Dot MKIII. No real noticeable difference.
The Antlion ModMic isn't a desktop microphone; you attach it to your headphones. If you're OCD about adhering something to your headphones, I don't recommend it.
I didn't include any closed headphones because it's not my area of expertise in terms of gaming. At least, not anything near your budget (Alpha Dogs are the only thing I think come to mind as a good closed for gaming, and they're double your budget
without a microphone and amp). Creative Aurvana Live and a Modmic/Snowflake might work.
A lot of gaming headsets are open; really, the only ones I recommend are. I haven't found a good closed one. Not to say all open headsets are good (lol Kraken Pro).
Good rule of thumb: if people who have heard it commented that it has a large soundstage, it'll probably do good for gaming. This is all assuming you use software like Razer Surround, Dolby Headphone, etc. Or just turn on binaural audio in your game's settings. This was written assuming you PC game; if you don't, the headphones are all still good recommendations.