There is a shotgun for every conceivable situation. Do not get overwhelmed. In short, shotguns come in two styles, Magazine-fed and break action. For shooting sports, both are acceptable.
In the magazine-fed style, the semi-auto, also know as the autoloader, is preferred (third gun pictured). These are easy to use and allow for two shots without the user needing to load the second round. Semi-auto shotguns are also very easy to find, and in most cases, they can be less expensive. Prices vary, but quality semi-auto that will take the punishment of shotgun sport will run in the neighborhood of $1300 to $1800. They do have lots of parts that need cleaning and oiling regularly. Recoil will be much less with an auto-loader, which makes it ideal for smaller-framed youth. Most youths start with an auto-loader before progressing to break actions like over-under.
Break action shotguns. The most common of this style will be the over-under shotgun (bottom gun pictured). Over-under shotguns are also easy to use and allow for only two shells to be loaded at a time. These shotguns can be heavy due to their construction. Additionally, over-under shotguns have greater felt recoil. This means the user will feel the recoil much more in an over-under than in a semi-auto. They have very few moving parts, but still require cleaning and regular oiling. Shotguns in this style are more expensive when comparing them to semi-autos of the same quality and grade. Over-under shotguns can run you from $2200 to $30,000. Please do not buy a $30,000 shotgun. Entry-level over-under shotguns can be found for less than $1000, but these models typically will not withstand the test of time when it comes to shotgun sports.
Before you buy any shotgun, I would strongly encourage your youth to try one out at practice. Many of the older athletes have guns that your athlete can try out. See if it is a good fit for them. It is also worth noting that gun fit is important and that most shotguns will not fit your athlete right out of the box. Talk to the coaching staff about gun fit.
When buying a shotgun, you might be asked a ton of questions from the sales rep. Remember, they are there to sell guns. Knowing the following basics will help you navigate the buying process.
Keep it simple. You do not need any fancy recoil reduction technology—for example, Beretta's Kickoff system. Just go for a gun with a solid wood stock.
Barrel length is not as important as you may be told. A 28-inch barrel is perfectly fine for all shotgun sports.
Weight is important. Guns that are too heavy are hard to make lighter. Guns that are lite can be made heavier.
If your child continues with these sports, this will not be the last shotgun you buy. Beretta, Benelli, and Remington semi-autos are high-grade guns and can last a lifetime. Tristar, Mossberg, and Franchi are good mid-grade guns that have a good reputation for being very rugged and reliable. These mid-grade guns are not as refined as the high-grade guns and will tend to lack some features and or adjustability.