Unrelated to the actual question but related to the title: Check if the controller fits your hand size. For example, I don't like the Xbox controller and much prefer the ps4 ones since they fit in my small hands better.

Consoles have used different wireless tech/protocols but mostly use bluetooth now. Bluetooth devices should be able to work on pc but may require extra hassle like non standard protocol and no official drivers if they weren't intended to be used that way. For the ones that do work the main takeaway is no additional dongle if your system already has bluetooth, but there may be addititonal input latency.

2.4ghz is kinda silly naming because bluetooth and wifi are both 2.4ghz, though wifi also has 5ghz, but it usually refers to presumably proprietary protocol over 2.4ghz and uses a usb dongle. It will generally be faster/lower latency but that isn't even guaranteed as the 2.4ghz mentioned never refers to any standard.

The latency difference will only be noticible to cats or birds or something unless the controller tries to pass the headset audio to the pc. Bluetooth audio has a noticable delay compared to most '2.4ghz' options, particularly noticable in rhythm games, games with inputs timed to audio cues, voice chats where everyone else has low latency audio and fast internet, and probably other stuff I'm not personally into.

Deciding your controller comes down to a lot of things. If the means exist in your area, try as many as you can to determine your most comfortable size, shape and layout. Other than that you absolutely must do yourself a favour and get a controller with hall effect sticks. The console companies all love stick drift. Don't be fooled by Sony's replacable stick controller either, the sticks are always out of stock and if you can solder you can get a regular controller and swap to hall effect yourself, both sticks, when drift starts for like $5 CAD vs $25 for 1 that will eventually die in the same way.

But probably just don't give m$ sony or nint your money since they all get drift and I have replaced the battery and charge circuit on 8 separate ps4 controllers and drifting sticks on so so many of every big3 console controller. 8bitdo has a few stick and trigger hall effect controllers which I can mostly recommend, I have the hall stick only version of the ultimate controller and several of their older regular stick controllers. They are noted for their very close replica feel of snes dpad and face buttons but the configuration and firmware update software was windows only last I checked. I also quite like retro fighters controllers but all of their hall effect options are on closed preorder pending fulfillment. There are other highly regarded non big3 brands that I haven't tried that seem to have pretty solid feature set and build quality too.

The used market is also an option. Arrange for testing though, and pass or haggle on drift, charging problems, filth, etc. Ifixit teardown guides for swapping parts or just cleaning the shells and button caps in soapy water with a soft rag. Watch out for devices originally sold with 'soft touch'/'smooth grip'/etc finish as all of them have degraded and become sticky even unopened in box by now.

For Linux, I recommend the DualShock 4 (PS4) and DualSense (PS5) controllers. They have native support built into the kernel, so you don't need to install drivers. They're great in Steam, emulators, Wine, and most native linux games. They work in both USB and bluetooth mode. Motion controls work. Touchpad works. Rubmle works. Dead zones are nice and small.

The only features I'm not sure about are the DualSense haptics and adaptive trigger feedback. There was work happening on those when I last looked a couple years ago; I haven't checked recently.

A few people have reported lag with certain bluetooth adapters. I haven't seen it with any of the hardware I've used, but if you encounter it, you can always get a different bluetooth adapter or exchange the controller for some other model.

Never buy a PS4/5 or switch controller they are all overpriced garbage they intentionally make for ewaste landfills in 2-4 years.

I really like my gulikit king Kong 2 and they made a elite controller style one with the 3rd version so I'd recommend that. They're hall effect joysticks which feel real nice and are easy to repair and customize.

I've been really happy with my gamesir g7 se.

I game on PC every day, and I always use a controller. I've tried many different controllers over the years, and most of them have had issues right out of the box. Only one controller has been perfect. The razer wolverine v2.

https://www.razer.com/ca-en/console-controllers/razer-wolverine-v2

It is super well built, and has a nice heavy feel that feels great if your hands are slightly larger than average. The thumb sticks are very grippy, have uniform resistance in every direction so you can be super accurate, and have zero stick drift after years of intense gaming. The face buttons have a mouse click feel to them, which I love. I always hated the face buttons on other controllers, including the genuine Xbox Controller, because they felt mushy and have inconsistent press and release.

I've tried 8bitdo, they felt cheap and uncomfortable. I've tried the Xbox Elite Controller, but it had tons of quality issues. I haven't tried Scuf, so I can't comment on that. But this razer controller is by far the best. Once you've used these clicky face buttons, you'll never want to go back to a normal controller.

I highly highly highly recommend it. The only downside is that it's wired, but they have a more expensive version that is wireless.

ignore brands whose identities amount to "RGB lights, edgy packaging, and GAMER in big letters", they're overpriced shit 100% of the time.

Flydigi Vader 3/4 Pro. Been using one for a couple months. Hands down best controller I have used. Hall effects, trigger locks, rear buttons, mechanical face buttons, glorious d pad. The not so fun is the crap software and joystick defaults. Once you update on pc it is great. Under $100 for either model.