For Sporting Clays an auto-loader will work better. Less recoil too. JMHO
The less recoil thing is a consideration and is certainly an advantage. However, my preference in sporting clays is for a double (over/under) of some flavor because of the simple fact that you have two chokes instantly available. Any good clays course will at some point force you to make a long shot (35+ yards) on the same station you have to make a short one (20 or less yards). With any autoloader, you only have one choke available and it will likely be a compromise choke for both of those shots. That said, there are those who shoot clays at a very high level with autoloaders. It can be overcome. If recoil is that big a concern with an over/under look into things like porting and recoil compensators.
Next up. Gauge selection. If sporting clays is the only game you're going to play, you'd be fine with a 20 ga recreationally. As soon as you start shooting registered targets/competing however, you are putting yourself at a significant disadvantage with anything other than a 12ga. There are sub-gauge classes at bigger shoots...but that alone should tell you something. Additionally, if trap is on the menu as well, there is no choice. Skeet ultimately requires that you shoot all 4 major gauges (12, 20, 28 and 410) so no big deal there. Long story short, the 12 gives you far and away the most versatility. Additionally, ammunition is significantly cheaper for the 12. While this seems counter-intuitive, it's really just a volume game and 12 is where the volume is. Don't underestimate this factor as the amount of ammo you can burn through in these games is staggering. In 2006 my wife and I combined to pull the trigger on trap targets about 21000 times. It gets spendy.
Finally, as to guns for specific games...The biggest difference is trap. Trap targets are relatively fast moving, long range (usually broken in the 40-50+ yard range) and always rising. The guns are built to match the game. They shoot over the point of aim pretty dramatically. Patterns are usually 80/20 or even 90/10 over POA. Makes sense if you think about it since to lead a rising target with a gun that shoots at POA you have to cover the target. Chokes are tight since the range is long. Barrels also tend to be very long. 32" is the norm but 34" is not unusual. They are ALWAYS 12ga and usually a combo. (Set of two barrels, one single one over/under to allow you to shoot the same gun for singles trap, handicap trap and doubles).
Skeet...really, just sporting guns that get optimized for target shooting with the following: 1) longer barrels since brush etc...will never be a factor. Typically 30" +\-. 2) slightly more weight, to breed a more smooth swing. 3) Usually extended choke tubes or backboring to lead to more uniform patterns. Like all target shotguns they will often employ adjustable stocks. Combos again, are very common here since at higher levels, skeet shooters will shoot all gauges.
Sporting Clays- generally, the most similar to field guns. Typically, the things a good intro-intermediate clays gun will have that a sporting model won't? Porting/recoil compensation, extended chokes/backboring and an adjustable stock. Clays, until you really get competitive is very much a "run what you brung" game. You CAN shoot it with anything other than a single shot...the info I gave is for those looking to be competitive in leagues and/or beyond.
Phew. Questions?
