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I like to go with 12 because you get more shot per shell, and therefore a better chance to hit the target. When skeet shooting, you are most likely using #7 1/2 or #8 size shot, which is usually a low brass round that doesn't have as much kick as, say buckshot or a slug. Some people like to use a 20 gauge because that is what they are used to and it has slightly less recoil. Ammo is also a bit cheaper. Ultimately, you should try both and go with whatever feels better to you. If you get better results with a 20, stay with that.

As for chokes, something like Improved Cyl. or wider is best to get the biggest coverage to hit the target. You don't need much to break a clay, so 1 or 2 pellets could very well do it by themselves.

Just go with a 12 gauge. Ammo can be found pretty cheap. Gauge doesn't matter too much.

Also; don't listen to people who say you need an expensive shotgun to do well. Last time I went I got, out of 25, 21 and 16/17 (I forget which). I just use a standard 870, and that works just fine.

Before buying ammo, check the rules of the range you plan on going to first. Ours only let's us use 7, 7.5, or 8 shot, 1 oz maximum, at 1200 fps or less.

That would be awful for skeet shooting and you know it.

DRAGONSBREATH.

Yeah, I know. Clay is non-flammable, and the burning pellets aren't enough to crack a clay, BUT IT WOULD BE AWESOME IF IT WERE.

@OP: Last time I used a 12-gauge (Beretta Xtrema, IIRC) I dusted the entire course (50+). Recoil on such small shot is less than a .30 cal, and really, with correct timing and good aim, you could shoot skeet with buckshot. 12, 20, whatever feels good to you. Personally I'd use 12, because my gun would be multi-purpose, and I'd want 12 for the other uses. Like home defense.

no problem

Whatever you can shoot the best!

I've watched people shoot damn good with .410s, even if they were just showing off to everyone. It's all about how good you can shoot with that shotgun! I usually use a 20 gauge side-by-side, an old Savage workhorse. I've shot a couple rounds with borrowed 870s (all 18 1/2 Cylinder bore). Eventually, I'll get around to trying it with my 20" cylinder-bore M500.

What KIND of skeet? There are several kinds. Just the stereotypical casual-style 25-clay, single-station, you should be fine with pretty much everything. But you get into this 150+ round multi-FIELD stuff, you'll need some serious gear. I'm going ot assume it's more casual style. Grab a shotgun that, above EVERYTHING ELSE, FITS YOU BEST.
I cannot strain this enough. FIT IS EVERYTHING. Proper fit will allow it to just "point" naturally, without requiring actual aiming. It will reduce how much it beats you, and it will make it swing a lot better.

Nice knowing you!