I just started myself. Some of the tips I've gotten that have helped...
Get a smooth motion mounting the gun to your shoulder. Get your cheek on the stock so your eye is over the barrel. Make sure your dominant eye matches your dominant hand, or adjust by shooting offhanded, or with one of the mechanical options available...cross over stocks, sight blinders and so on.
When you get to the gun club and they'll have trap houses- Point your front foot at the trap house, have your back for about 45 degrees. Point your muzzel just above the roof of the trap house and cheated just a smidge to the side of the house you're on. Swing through. Once you get your bead on the bird, don't stop and try and refine your aim... shoot and follow.
Most of it probably is being new. I was and still am new. Most people who have given my pointers talk about an Aha! moment, where it clicks and you'll see what people are talking about with pointing-not-aiming, and follow through and what not. I recently had my first moment in that one. So keep going, grit your teeth at your 8's and 9's. Fix one thing at a time. My Aha! moment came from bying one of those rare-earth-magnet hi-viz lightpipe sight things. All of a sudden I could see both, focusing on the bird and still see the "sight bead" of the light pipe in my peripheral vision
Use the right tool for the job. I wouldn't hang shingles with a 3 pound sledge. One of te last times I was out, there was a guy taking his first round of trap with a Saiga home defense. He sufferd through as well. Part of that, and I just realized noone has asked you this, what size shot were you using? For clay shooting, the Walmart specials, 7 1/2 or 8 are good practice rounds.
My uncle got me started on this hobby. He's got a trap range and pheasant preserve on his farm, and on a visit he took my brother and I out to the trap range, loaned us an over-under and had us go to town. When I got back I started watching the local used gunstores-maskquerading-as-pawn-shops and Cabelas Gun library. I found a really nice Citori field grade 30 inch for an acceptable price tag. I plan on picking up a new Browning BPS soon as well.
You may want to pick up a BPS for a (relatively) cheap longer barrel shotgun. Though, depending on your inteded uses, your "ideal" gun could be different as well. Over/Under are extremely popular. And usually priced to reflect that. I would liken them to BMWs and Mercedes. Pumps(and Semi-auto's, in this analogy, would be Toyotas. Side By Sides would closer to the classic cars. Model T's, or 1929 Chevrolet Roadsters. Very popular a while ago, still a fine working machine, fully capable of being souped up, but they look different, feel different, and have to some extent (at least the shotguns) fallen out of favor.
If you have an interest in hunting, I'd take different guns to different hunts-
Ducks/Waterfowl, or Turkey, I'd go with a pump or SA. Sythetic stocks/finishes can come in a camo pattern, have fewer issues with the water involved in waterfowl, and are just plain cheaper if you drop it in the lake.
Uplands Bird hunting your pump/SA will work, but I'd also start looking at O/U or Side by Side. I wouldn't get a SxS myself for continuity reasons, but your mileage may vary.
If all you want to do is shoot clays, I'd start with a cheap but still quality Pump or Semi-Auto first to decide if you really really enjoy the hobby, then I'd look into getting an Over/Under. Some of those are almost a trophy in and of themselves and can run thousands and thousands.
The brands/Models I've heard the best of are the Browning BPS, the Remington 870 Wingmaster(Not the express), and Weatherby's.