Where does one start if they want to compete in a sport representing the Army?

The program you should be looking at is the All Army Sports Team, specifically the wrestling team*. All COMPOs of the Army are available to apply.

Details for the process, qualification, etc. are on the webpage.
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* All Army Sports Team (Wrestling) - https://www.armymwr.com/allarmysports/all-army-sports-selection-criteria/wrestling-1

I sent one of my lieutenants to the World Class Athlete Program, but you have to already be just this side of qualifying for the Olympics. https://www.armywcap.com/apply That doesn't sound like you. As a SFC in the Guard, you're probably too old to be competitive even in interservice sports... it's a young man's game!

For something like wrestling, the competition for WCAP starts with kids that placed at state in HS, and/or placed in at least conference in college, and is typically coming right off that last competition year.
But for WCAP in any sport, that's pretty much your job much of the year in the Army, and it is open to all three compos (USA, USAR, ARNG).

I have also had friends in the Guard who volunteered to assist with local high school wrestling, with the school during the season, and most of the team during club/freestyle season, and got a lot of wrestling in that way.