You'll need a raft and paddles (usually 8 people per raft) to navigate the rapids, as well as a good life jacket and helmet. If you're heading out into cooler temperatures, wear warm, waterproof clothing and wool socks—these will keep you warm and dry while splashing down the river. If it's warm outside, nylon shorts, a bathing suit or swim trunks, a t-shirt, or a tank top are all good choices—just make sure to lather on that sunscreen to protect yourself from the sun. As for shoes, your best bet is to wear a pair with rubber soles or slip-on water shoes that you don't mind getting wet—because even during a good run, you're sure to get wet!
Some rafting trips can take a few hours (or even an entire day), so pack plenty of drinking water and food—just make sure to pack it up tight to keep the water out.