What's not to like about clay target shooting? Break lots of things, make all the noise you want, when you were a kid you would get in trouble for doing those things. For a beginner to start from scratch and looking for a first gun follow the previously shared suggestions. Just to add - if you are average size most average shotguns will fit well enough to enable you to hit some targets and enjoy it. Most field guns meet these specs, autos and over/unders are easier to manipulate for this purpose but a good fitting pump with at least 26" of barrel (preferably longer) will work fine to learn the basics of 16 yard trap, much more difficult for learning skeet or sporting clays because many of these presentations require two very quick shots.
There are a number of new autoloaders within your budget, mostly made in Turkey, some good, some bad, many variable for reliability. If you decide to go this route concentrate on known brands with a positive track record such as Savage/Stevens, Weatherby, Stoeger, CZ, Huglu, etc. These companies have a reputation to protect , make a decent product and back it up. Do your research, easy to do on line.
A good alternative is a used over/under or auto by one of the well known major manufacturers. Look for Beretta, Browning, Ithaca/SKB, Remington, Winchester, etc. As a beginner you will be best served by a Modiified choke so at least one barrel should have this constriction, either factory or gunsmith modified, and removable choke tubes will also have this capability although not necessary. Many older over/unders are choked Modified and Full, great for casual trap and sometimes available near your budget because fixed chokes aren't " fashionable" now. Original fixed Modified choke barrels are scarce on autoloaders and fetch a premium but a Full choke barrel is easily opened to Modified by most competent gunsmiths for less than $100.
If you enjoy the sport and can see encouraging improvement you will almost certainly upgrade away from your starting gun after a couple of years or less but keep it simple and fun to start with. Have fun, break 'em all. J