All good points for you above.
I LOVE clays, but have yet to set foot in a competition. It's a great lifelong sport, and it's great news that your wife love it too!!! There's a beauty .410 citori in the EE if you have little ones, or, if you're like me and enjoy embarrassing people with big mouths by beating them in a round with a .410 while they have a 12g!!! I'd be buying it myself, but we already have one. We'll knock off 1000 rounds easy with friends in our pasture on a weekend.
I agree fully with the "stick to 12g" comment. We bought a citori 20g for Megan, and it kicks just as hard as my 12g because of the 1.5lbs in lost weight. The only advantage of the 20g is that it's a bit easier for her to swing.
As for length of barrel, get what you can, meaning don't pass on a good deal because the barrels are 2" from what someone said is "best". Being fitted is crucial. I'd suggest looking for a gun with a raised rib and barrel ports. If you find one that doesn't quite fit your face and build, there's a recent thread on here of someone who had a adjustable comb added for 185$. Don't be scared of used. Most people that buy a citori or perazzi or beretta also take care of them. I don't know much about semi autos, so I'm not of much help there. I do know the versa max and benellis are highly regarded and quite good value. I love our o/u's and see them as really classic heirlooms for our family, but that's just me. You may feel the same about a semi.
These guns are not common, but keep the money set aside, cause occasionally you'll find one on clearance. My trap XT was 1900$, and the .410 was 1600$. That's when you should jump on it!! If cash isn't a huge issue for you, there is a gorgeous XT in the EE right now, but it's pretty fancy and around $4k.
When you do find a gun, don't cheap out on the case!! Browning has some beauty leather and waxed canvas ones that are reasonably priced on optics planet.
I'd also suggest, at the risk of being taunted/bashed/poked at by various CGN heroes, a recoil shirt. No matter how thick you are, 250 rounds of 12g is gonna leave a mark. We both have Evo shirts, and love them. No bruise since I got mine. About 55$. They are meant to fit tight like an under armour, and you wear whatever overtop.
If you're hoping to do both trap and skeet with one gun, it'll always be a compromise on both. For skeet you want a very snappy little gun, 26"-ish barrel, and for trap you want that smooth swing and added pellet velocity of the 32"-ish. I'm a fan of the two bead sights, and the hi-viz bead at the muzzle. I'd honestly pick one for now, and if you want to go to the other later get a second shotgun, but that's just me - compromise bugs me.
As for why some shooters use one barrel vs two, and short vs long, etc - there a lots of variations of this game. Single, double, American, Olympic, etc, etc... I don't know them all, or what the rules are, but I know they're out there and that's why there are so many different variation on guns, and opinions to follow. I do know that to be competitive, your looking at being in the 97-99% hit rate, sometimes even 100%. Don't let that discourage you, but it's true. Around here I know that it's also an "old mans game" - meaning there are few young shooters. My contact at our club is my age (37), and he says he's the youngest by 20 years. It's sure be nice to add some more 12-50 year olds!!! When I was a kid it was a "rich mans game", but now that a pair of skates cost 600$+, and a hockey stick is 300$+, I'd say it's now pretty average....
I only have 2 technique suggestions. 1. IMO your shoulder and cheek weld are crucial to getting your eye looking right down the rail. Hence the need to fitting. Practice bringing the gun from your hip to your shoulder. 100reps per night to develop muscle memory. Only takes 10 min, and helps a lot.
2. Clays are not a game of precision, they're a game of feel and flow. They draw on the deeper, more "automatic" parts of your brain, the ones that make your slapshot 10MPH faster, or allow you to hit home runs. It's 85% hand eye, and 15% muscle. Do not overthink, let it flow. Take a deep breath, shake things loose, and enjoy!!
Wish you all the best!! It's a damn fun game!!!
-J.
ps - after you get a few 10/10 or 25/25, let me know and I'll send you the rules for 20/10!!!!