I have a set of the Vmoda Crossfade Wireless 2 Codex Edition (a mouthful of a name), and I really like them. They offer great midrange reproduction and a good overall soundstage, especially for their totally closed design. With bluetooth capability and support for lossless streaming, it's a really competitive model with a lot to offer, though it comes in around $350. The Vmoda m100 are wired-only headphones that are otherwise pretty similar to the Codex, with a slightly more aggressive bass curve. They come in under $300.

In general, Sennheiser makes outstanding options at nearly every price range, so I'd encourage looking into some of those. Off-hand, the HD600 is around 300 but it does not offer bluetooth.

the Sennheiser HD 6XX on massdrop are amazing, so there's that. they're only 200 USD

If you travel a lot and would find some noise cancellation useful, I'd fully recommend the WH1000XM3's.

First off, they are wireless which to you seems to be a plus.

Secondly, they have extremely good battery life and fast charging, 5 minutes of charging will give 4 - 5 hours of playback. Of course, this isn't linear but it is incredibly useful.

Sound quality is great although I'm no audiophile. I do find them to be a bit bassy but you can quite easily equalise it if you wish to.

Noise cancelling is the best in class and the main reason most people get these headphones, it will block out all noise which is great when travelling. If you find yourself in a situation where you want to hear things around you, for example: if you are gaming and want to hear if anyone calls you, you can flip the noise cancelling around and project the outside world into your headphones.

I guess another thing is they have surprisingly good surround sound capabilities which, when configured correctly, is great for directional sound in games.

They tend to hover around the $300 so this is the edge of your budget.

If you want a decent headset, with a good mic. Either the Astro A10's or the Astro A40's. Good headset for just about anything.