Indiana's rivers and streams provide a great way to experience the outdoors, although the canoeing or kayaking afficianado will find very few real challenges. Mostly, canoeing in the Hoosier State is just a fun way to spend the day, not too strenuous, and a good way to get in a lot of sightseeing, birdwatching or wildlife viewing.
There are, however, at least two trips that can prove to be fun even for those looking for a little adventure. The Big Pine Creek in Warren County, from Rainsville to Twin Bridges (near the Fall Creek Nature Preserve) and from there south to Attica, consists, when the water is high enough, of a series of mild to exciting rapids. The water is only high enough after really heavy rains, and even then usually only in the early to mid spring.
The Whitewater River is also supposed to be pretty interesting, but I haven't done that one yet myself. I have canoed Tippecanoe and Wabash Rivers, Wildcat Creek and Sugar Creek (the one by Shades and Turkey Run State Parks) and can recommend all of them for one reason or another. The big rivers are usually fine (except at flood stage) for the not-so-experienced paddler.
One interesting resource for paddlers is the USGS real-time monitoring of stream flow, found at this site. You can click on any of the monitoring stations and find the height and flow of a particular stream or river. With some experience, you'll be able to tell what kind of experience you'll have; for example, a good height on the Tippecanoe is about 2.4 feet. Too much lower, and it's all bumps and drags. Above 3 feet, it loses a lot of character.
As for the rest of the rivers and streams and for information about put-in and take-out points, there is no point in redoing something that someone else has already done very well: click here to link to the Indiana DNR's Indiana Canoeing Guide.
My own personal trip reports
Big Pine Creek