Get in an air filled raft and head down the river. You have a raft captain that needs to know what he is doing, but you can be clueless. It's that simple. The one thing you have to do is pick how strong a river you want. There are six main levels (sometimes they use a - or + to indicate weaker/harder versions of the six main levels).
Class I, is mild, Class V is harder. Even the youngest child can do a Class I or II. Teenagers should be able to handle Class III at the very least.
But if you are looking for excitement, then you want Class IV or Class V. Don't try class V until you have had some experience on a class IV.
But wait, I said six levels, and V stands for five. So yes, there is a Class VI. It is the term for a river that that is too harsh to run. For example, Niagra falls is a Class VI rapid. Do some people go over it anyway? Yes. We call them suicidal. Don't try this unless you are a) an experienced raft captain and b) not taking anyone else with you.
I have done Class IV rapids in Alaska. It was a lot of fun, the wet suit kept me warm and dry. I do like speed.
Note, price depend a lot on length of the trip as well as the class. It's only cheap if you go for a short 4 hour or shorter trip. If you go for much longer than that, it edges into expensive, especially if you are talking class IV or V. If you go for a full on multi-day camping trip, expect to pay thousands. While it takes a lot of skill to captain the raft, you need no training to just do it. While you don't have to be in good shape to do it, you should at the very least be able to swim. If you can't swim, stick to Class I or II.
You will have to wear a life jacket even if you can swim. A helmet is heavily recommended. In colder locations, a wet suit (including shoes) is needed.
If you want a Class IV or V experience, you may be dependent on timing - rivers run faster when dams let water out or when snow melts.
White Water Rafting is gender neutral. They take kids as low as 4 years old (at least for the milder stuff). Seniors are welcome, especially for the milder runs.
You can learn more about White Water Rafting here: