Ignore everything that has been said and start from what is really important and what will make you a good fighting game player regardless of the game: the fundamentals.
You have to learn the very basics which are the backbone of every single fighting game and I'd rate above everything the concept of spacing.
You'd be really surprised by how many people I've beaten simply by doing 2-3 moves with a character and winning several rounds because my opponents were big on combos and executing special moves but they couldn't lend a single one on me because I'd always stay at the correct distance to punish them every. single. time.
New games have generated an absurdly amount of scrubs which think "hurr durr, I'll just get the best internet connection and fightstick and practice 24/7 my combos and 1-frame links and I'll be godlike" and then spend all of their times on reddit saying how the game sucks and how unfair certain players are.
So just pick whatever fighter you like (Street Fighter II would be perfect for this, but since we're talking Neo Geo, any KoF or Fatal Fury will do, although don't pick the ones with planes swap) and look for some tutorial online which will explain the concept of being always at the right distance from your opponent so that you're not too close to get hit but not too far so that you're unable to punish whiffed moves.
While going through this, start researching mechanics like hit-stun (how much time you're unable to react after you've been hit), block-stun (how much time you're stuck blocking after you've blocked a hit) and recovery (how much time you need after attacking to be able to block) and start-up frames (how much time an attack takes to happen).
You'll see that there are some very quick moves which have minimal start-up and recovery, but they cause very low hit and block-stun as well (which means, for the hitting part, that you won't be able after a successful hit to follow up with another move, thus increasing the damage dealt, this is the basic for combos, and for the blocking part, that your opponent will be able to counter-attack or do something else quickly after).
Once you've mastered this concepts you're already a *VERY* good player and you'll be able to make several people rage quit just by controlling space properly.
Going forward, you should start researching the game's specific mechanics and that's another whole story and depends heavily on the game.
Get a good stick, but don't go overboard with it or, if you've always played on a pad for your entire life, KEEP PLAYING ON THAT FUCKING CONTROLLER.
Holy shit, you have no idea how many players force themselves to learn playing on a stick and end up being absolutely garbage while they are already good with their own pad.
If you look at big tournaments you'll see that there are several top players playing on a pad, so just play with what you know best and feel most comfortable with. If you want to try out a stick, then sure, go for it, but don't force yourself down that path.
And one last thing: fighting games are hard.
Don't expect becoming the new Umehara in one month, it takes YEARS of practice and experience to become a good player, so don't give up after a short while, keep at it, practice and play against other people and ASK THEM for tips and advices and eventually you'll get there.
EDIT: some good reads you should definitely go through:
http://shoryuken.com/2008/07/21/domination-101-so-you-want-to-be-a-dominator/
http://shoryuken.com/2013/07/24/domination-101-controlling-space/
http://shoryuken.com/2003/07/21/domination-101-on-cheapness/
...and all the other articles by Seth Killian.