Different clubs have their rules regarding safety - if you aren't sure, ASK. Safety first.
Trap is different from skeet which is different from 5-stand and sporting clays.
Basic trap involves a single target from the 16 yard line. Skeet involves singles and doubles as you move around a semi-circle making the presentations different. In both cases, the fields are set and you will find that each one is the same from Maine to Florida to Washington.
5-stand is unique comparatively so as each one is different. Anywhere from 8 machines and up can be used to give a variety of presentations. Typically you'll have a single and two pairs at each stand - sometimes report pairs, sometimes true pairs.
For skeet or 5-stand you WILL need a gun capable of shooting two shots. While a pump is doable, a semi or O/U tend to rule those roosts due to the ease of getting off that second shot while on target. Guns can range from a few hundred dollars to well over $100,000. FIT is what really counts when shooting moving targets. A pump that truly fits is better than an O/U that doesn't.
When you go to the local club, ask for help. Ask to borrow or rent as many types of guns as you can to see what fits and feels right to YOU. What works for me may or may not work for you.
Whatever type of gun you decide on - do yourself a favor and buy a decent one. Quality guns last longer, resell better, and have a tendency to do the job better. Look at the good shooters at your club, see what they use, ask to try them, get a feel for what you want, then buy once and enjoy
Good luck!