2 things to look into.
1. NRA. A great number of legit, serious, and responsible gun clubs register with the NRA. This helps ensure a degree of attention to safety and training standards that are at least within shouting distance of what everyone considers "normal". Not only that, the NRA is associated with a whole range of shooting competitions suitable for almost anyone with an interest, from plinking with pellets (way way harder than it sounds) to hardcore heavy load blasting.
2. Local conservation or wildlife groups, surprisingly enough, often own shooting ranges. Especially in areas where hunting is a normal sort of thing to do, there may be local groups who own ranges that are used for instruction and competition by all the shooting clubs and organizations in the area.
In any case, you probably want to look for gun ranges that are used for competitions and find out what organizations sponsor and sanction those competitions, and get in with the crowd that interests you the most. I shot standard pistol and air pistol in college, and was planning on continuing via NRA events until I got a real job and ran out of time. If you're really good and are lucky enough to live in the right area (colorado springs for example), you might get lucky enough to shoot at a USOC (US olympic committee) run facility, with access to their instructors. My instructor was an olympic bronze medalist.
