Basically you can do whatever you want, whatever works for you.
Maybe best to start with your phone? Is it unlocked? Will it work in Thailand? Will it roam in Thailand? Do you want to be able to receive calls made to your U.S. number?
You can purchase a SIM at the airport, or at any one of the billion mobile phone shops/conveninece stores. Maybe purchase one at the airport if the wait is not too long?
Standard SIMs are 49/50 baht, then you add value (top-up), then subscribe to a mobile data plan. Tourist SIMs typically have bundled voice minutes along with data, and are sold as a SIM with the service.
Maybe if you shared details re: your phone/requirements we could make better recommendations?