Wake and Johnston County are very child friendly with a ton of things to do! Along with YMCA, the area Parks and Recreation departments have a lot as well. (Do a search for Raleigh Parks and Recreation or Cary Parks and Recreation, Garner parks and recreation, Apex parks and recreation, clayton parks and recreation) That should get you started.
Here are some links of a few area places that offer leagues or classes for kids in the Claytin or Cleveland area of Johnston County.
Greater Cleveland Athletic Assoiciation- offers baseball, softball,basketball, volleyball, and soccer teams.
www.gcaasports.com
Clayton Little League-
Clayton Little League
Morgans Gymnastics Gym-
Morgan's Gymnastics Academy | Smithfield Clayton Garner Raleigh NC North Carolina
Premier Athletics- Cheerleading and dance
Premier Athletics - Clayton
Kindermusik- teaches music to children
www.kindermusikofclayton.com
Pop Warner of Cleveland- Football, cheerleading, and Dance teams for kids
Cleveland Pop Warner Association - (Garner, NC) - powered by LeagueLineup.com
That should get you started with a few things to consider for Johnston County. Like I mentioned before, the Triangle is family friendly. You will find much more when you narrow down what area you are considering. There are teams, gyms, dance studios in every area.
Good luck!