Actually putting your hands on a controller and playing the game can be a challenge in and of itself. Here's a few options I've tried over the years and some impressions:
Controllers First thing to note about using controllers in this game is posture. I tend to grip the pad while bracing the controller on my knee so I can use the buttons claw-style. You gotta hold up to three specific buttons at a time, so elbows-on-desk or freestyle is no-go.
Standard Xbox One - this has been my go-to for casual emulation. I like that the d-pad is microswitched, and the buttons are membrane. The buttons are a bit small for playing with your right-hand claw style, but it gets the job done.
Hori Fighting Commander (non-Octa) - The PS4 variant has a slightly more comfortable dpad than the stiffer Xbox version. In either case, the pad and buttons are both membrane. Having large buttons is a plus, and there's generally enough room for everything. A membrane pad means it will wear out faster than switched pads, so longevity is an issue. In the meantime while it's in good condition, though, it's a darn good option.
Hori Fighting Commander Octa - This newer variant of the pad uses microswitched buttons and a redesigned pad. And I hate it. The buttons feel great, but switched buttons in Rage means you can accidentally press too early. The dpad has sharp corners and a slippery surface. That would be enough for me to turn it down, but the corners are legitimately awful. You can adjust their sensitivity in software, but it's either too much or not enough. Works great for retro gaming, but as a fightpad skip it.
8bitdo NeoGeo Wireless - while an interesting controller with four large pushbuttons, I had a hard time adapting to using a thumbstick. It's entirely possible, and depending on character, maybe preferable. It's not easy to recommend, though.
Sega-style pads - Take your pick. Retro-bit, 8-bitdo's M30, Retro Fighters, etc. As it happens, I grew up on six-button Sega pads and Primal Rage, so it's absolutely do-able. The teensy X,Y,Z buttons are not as ergonomic as it was when I was 12, so your mileage may vary.
PS-Style pads - It doesn't matter if they're membrane or microswiched 3rd party controllers, I have a hard time using the relatively stiffer Playstation d-pads to any degree of accuracy. If you're a diehard PS player, it may be a non-issue, but it certainly was for me.
Arcade Joystick When it comes to arcade sticks, there's a LOT of personal preference involved. How much resistance you want from the lever and buttons, long or short throw joystick, and even joystick gate. Broadly speaking, arcade sticks are the most comfortable way for me to play the game. After all, it was designed as an arcade game - make sense, right?
Hori (hayabusa lever / buttons) - Not a bad option for off-the-shelf sticks. They carry square gates, and have responsive low-profile buttons. As heavy-handed as I am, I'm slapping the deck like it owes me money, but I suspect most other people will be fine with it.
Japanese parts (Sanwa) - There's a bit of variance here, but by-and-large most commercial sticks use Sanwa parts, so if you're okay with that, the rest comes down to preference of enclosure. By default, Sanwa's are too light for me. They have only 0.9 lbs of resistance, so they are a feather touch, but that resistance is very even from side to side. You can grab a 2lb or 4lb spring for some light modding, and I recommend at least the 2lb spring. Similarly, there's not a lot of tactile feedback in their buttons. They have a little bit of mush in the press and feels really "hollow" to me. Still, it's the "standard" and most people expect that particular brand in their sticks.
American parts (Suzo / IL) - The reason I'm hitting the deck so hard on other types of sticks is because I am used to American parts. These are the firmest of the bunch. By default, Industrias Lorenzo's Eurostick has a spring like a doorstop. It's less precise, but has a quick return to center. Suzo Happ's Competition stick is not as firm, but still firmer than most. Their buttons in both cases also have a fair amount of resistance, but the press is soooooo satisfying. The unfortunate thing about American parts is their relative unpopularity among stick manufacturers. The only retailer to have something ready-to-go is AllFightSticks. Anything else is DIY.
Korean parts - I have much less experience with Korean and Korean-style parts, but I have tried the Seimitsu Nobi lever. It's short throw, and has a good, solid feel to it. The bullet top wore on my hand though, so I swapped it out for an aluminum bat topper. Much better. I find that my inputs are much faster, so double-taps like Vertigo's specials are a breeze. It may require some adjustment to your muscle memory if you find yourself doing those directional inputs, a smidge too fast.