opening with the most variations.

What opening has the most variations?

Depends on how Generic you declare an opening.  Just saying the Sicilian has the most variations as opposed to say, the Sicilian Najdorf and its many variations, is pretty generic.

Going that level of Generic, I can say the Queen's Gambit definitely competes.  The Sicilian has, for example:

Classical Dragon
Yugoslav Attack
Levinfish Attack
Najdorf old main line
Najdorf English Attack
Najdorf Sozin Variation

etc etc

Well, I could say the same against the Queen's Gambit.  Keep in mind, if the Dragon is a Sicilian, and the Najdorf is a Sicilian, the Slav is a Queen's Gambit, the Semi-Slav is a Queen's Gambit, etc.

Declined, Orthodox
Declined, Tartakower
Declined, Lasker
Declined, Cambridge Springs
Declined, Vienna Variation
Declined, Ragozin Variation
Declined, Rubinstein Variation (7.Qc2)
Declined, Exchange Variation
Slav, Central Attack, Bishop Sacrifice Variation
Slav, Central Attack, Modern Main Line
Slav, Dutch Variation
Slav, Exchange Variation
Slav, a6-Slav
Semi-Slav, Botvinnik Variation
Semi-Slav, Moscow Variation
Semi-Slav, Anti-Moscow Variation
Accepted, Central Variation (3.e4)
Accepted, Classical Variation (3.Nf3)
Accepted, 3.e3 Variation
Chigorin Defense
Albin Counter-Gambit

This is just the tip of the iceberg.  There are hundreds of others.

Question is too generic and pointless! 

The Ruy Lopez also has a ton. A quick glance at the wikipedia contents page showed 32 variations.

I think that probably would round out the top 3. For 1.e4, if black plays e5, there are a ton of ways to play the Ruy. 1...c5 (Sicilian) also has a ton of variations in its own right. For 1.d4, there really isn't any way to stop white from playing 2.c4 if they want to, so QG will have a ton depending on black's first responses (QGA, QGD, Nimzo Indian, King's Indian, all of which have their own variaations) 

If I were to learn one, which one do you recommend?

That depends on your goals. If you just play chess to eat away idle time, you don't really need to learn any of them. If your goal is to be a casually competitve player, then more than memorizing every variation of every opening, learn the goals behind the main variations of a few of them (you'll have to learn a few since you can't force your opponent to play what you want - if you learn the Ruy Lopez as white, but your opponent plays 1...c5 Ruy theory won't help you). If your goal is to become a master, you'll need to memorize several variations of several openings and have a deep understanding of the theory in all involved.

Like I said though, it mostly depends on what you want. I'm only 1255 on here in blitz, 1300+ live at 30 minutes, and just over 1200 USCF rated OTB. That seems low, since 2200+ is required for master. But in all of those categories, I rank above 70% or more of all other players, and I have no ambition to become a master. I have developed a basic understanding of the goals and theory of Ruy Lopez, Sicilian, English, Queen's Gambit, King's Indian Defense, and King's Gambit, but I haven't memorized all the possible variations of any of them (though I've probably played every major variation of the KG multiple times...since I've been playing it for 20+ years, but it's an opening that is designed to be sharp and give opportunities for tactical mate threats early [for both sides - a single slip up and black can pounce], it's main theory is avoiding the early diagonal attack from the vacated f2 square and getting your rook on the opened f-file).

Main thing is find something you like. If you keep playing you'll eventually develop familiarity with several openings, if for no other reason that you have to play against them (I couldn't play the Caro-Kann to save my life, but I know how I like to play against it because so many of my opponents have played it). Probably the first thing you'll discover is if you like fast-paced vs build-up, open vs closed. But the more you play, the more comfortable you will be in all of those, and you'll probably find your opinion changes based on both your color and your opponent. For example, I play totally opposite styles with white as opposed to black. When I'm white playing an opponent at my skill or lower, I favor open, sharp, fast-paced tactical games (love playing KG as white with equal or inferior competition). When I'm facing someone my skill or lower as black, I prefer to play slower, closed games and wait for them to screw up. But when I'm white and playing a stronger opponent, I will play more conservatively, such as maybe the Ruy Lopez, to try to give myself a very small advantage early and then see if I can carry it into an end game, since if I get into a battle of tactics with a superior opponent, I will probably lose quickly. On the other hand, if I'm black and facing a stronger opponent, I will abandon my closed game and waiting for his mistake before equalization because he probably won't make one that I'll see. So instead I try to mix it up as early as I can to see if I can throw him off, figuring I would lose anyway if I just let the game play itself out like normal.

sicilian!

Probably either 1. e4 or 1. d4, but maybe 1. c4 or 1. Nf3 as well.

It's called The English, and yes, with some obvious differences.