welcome to Cody, The boating is great here this time of year. Our local River, the Shoshone is dam released so we have had great flows all year and will continue to have them until the crops have been harvested in late September.
We recognize 5 distinct sections of river below the dam, with the top 3 ranging from class III to IV (+) the bottom 2 are known as the 6 mile and lower respectively and are mostly class II, but offer some good downriver play
There aren't many great park and play spots in our are mostly because the canyon is pretty deep and not all that fun to hike up or down, but if you ask a local they can still point you to some honeyholes.
The Clark'sfork is just north of us and a couple sections are still running, like Honeymoon and the lower CF.
The best way to hook up with boaters in the area our by getting on our message boards:
The Message Board
OR, the Cody Kayaking site on Facebook, which these days seems more popular.
I work at Gradient Sports, a local paddle shop and can steer you in the right direction as well, give a call when you pull into town: