Since you have a 870 also, you might look into "3 gun" if they have it in your area. Along with IDPA that Gunfyter mentioned, there is IPSC, USPSA (which are kind of the same) They have a class (production) that does not cost an arm and a leg to get into. You can shoot most any gun right out of the box. When you get into the higher classes like “open” that is where you can spend big bucks on “race” guns. You might also check with any ranges (or gun shops) in your area. Even some of the private ranges have matches that are open to the public.
Then there is bullseye, cowboy, benchrest, fast draw, high-power, pin-shooting, revolver, silhouette, small bore, tactical action, and a host of others.
Good luck, hope you find one in your area.
Remember, it is always safety first, then have FUN.