If your son/daughter enjoys completing, has had good success at the local level, and is possibly interested in playing at the college level, the next step would be to start competing in regional and national junior golf tournaments. Usually, each regional golf association holds an annual junior championship tournament, so that's one way to start competing at next level.
The national tournaments are run by independent junior golf organizations. Although most national tournaments are held in the Spring and Summer, some junior tours, like FCWT, schedule their tournaments starting in the fall months and continuing through May. The premier junior golf tour is the American Junior Golf Association (AJGA). And because there are so many good junior golfers that want to play in these tournaments, it is necessary to first establish a high national ranking to qualify for entry into these events by and playing and having success at other regional and national golf tournaments. Since each junior golf organization has different rules and regulations, it's best to review each association's website to better understand the tournament entry process and eligibility requirements. The AJGA website is quite extensive and is a great source for general junior golf tournament information as well. Links to national and regional junior golf tour websites are provided below.
National Junior Golf Tours
American Junior Golf Association (AJGA)
Future Collegians World Tour (FCWT)
The Laura Diaz FCWT Junior Girls Tour
International Junior Golf Tour (IJGT)
Callaway Golf PGA Junior Tour
Regional Junior Golf Tours
Adams Junior Golf Tour
Allstar Junior Golf Tour
Arrowhead Junior Golf Tour
California Junior Tour
Desert Junior Golf Tour
Florida Junior Tour
Florida Suncoast Junior Golf Tour
Hurricane Junior Golf Tour
Iowa Junior Tour
JAGS Junior Tour
Junior Players Tour
Legends Junior Tour
Midwest Junior Golf Tour
National Junior Golf Club
Northeast Junior Golf Tour
Northern California Junior Golf Tour
Northwest Junior Golf Tour
Oklahoma Junior Golf Tour
PARS Junior Golf
Plantations Junior Golf Tour
Premier Junior Tour
Robert Trent Jones Golf Trail Junior Tour
Southeastern Junior Golf Tour
Tarheel Jr Tour
Texas Junior Golf Tour
Top 50 Junior Tour
Triad Golf Today Junior Tour
US Challenge Cup
US Junior Golf Tour
In addition to the various junior golf tour organizations, there are a number of well-established, independent tournaments that also may be of interest. As with some of the junior tour events, entry into these independent junior tournaments is also highly competitive and are based upon a strong national ranking and in some case are by invitation only. Some of the larger national tournaments (e.g. USGA Junior), run local qualifying tournaments to qualify into the main event. Links to these other national tournament websites are listed below.
Junior Orange Bowl Golf Championship
Joanne Winter Arizona Silver Belle Championship (Girls only)
The Verizon Junior Heritage
Kathy Whitworth Invitational
Scott Robertson Memorial
The Bubba Conlee National Junior Golf Tournament
Southern Junior Championship
The Hudson Junior Invitational
The Western Junior Championship
Women's Western Junior
Junior World Golf Championships
U.S. Junior Amateur
U.S. Girls' Junior
Big "I" Junior Classic
Junior PGA Championship
Bobby Chapman Junior Invitational