A normal controller
Every game here works with an Xbox or PlayStation controller on PC, and Steam Input handles most others. The d-pad suits 2D games best; many players prefer it to the analogue stick for quarter-circle motions. If you already own a controller, start with it and see whether you stay with the game before spending on anything else.
An arcade stick
A stick with eight buttons copies the arcade layout. It is comfortable over long sessions and makes some motions easier, but it takes a few weeks to get used to. Look for one that works on PC (most do), and check that the buttons can be replaced if one wears out.
Leverless, or all-button
Leverless controllers replace the stick with four buttons for directions, like a keyboard. They are precise and popular at high level. Tournaments set rules for what happens when left and right are pressed together, so check the rules of any event you plan to enter. A keyboard works in a similar way and costs nothing extra.
Simplified controls, game by game
Several games now offer a second control scheme that puts specials and combos on single buttons. These are not cheats; they are full ways to play, usually with a little less damage in exchange for ease.