I went paddling down the Nenana River five or six times while I was working. It was my first time ever, and it was scary but fun. For a first trip, I'd recommend riding near the middle of the boat, on a guided trip, no paddling. If you are into a little (a lot) more adventure, sit in the front position on the boat, and grab a paddle. Going later in the summer gets you a better and more exciting ride, but early summer you get to see all of the wildflowers blooming.
I recommend the above company - they give you complete drysuits (meaning you wear your street clothes and don't get wet) and a heated bus ride back to the starting point (the river is cold).
If you go with a different company, make sure you get the drysuit. I think helmets are required now. If you are unsure where to go, ask the local workers. We know all.