Whats going on man,
I am a Hoosier myself but live out here in the four corners area and have gotten into boating since i moved west. My girlfriend is from Indy so we go home every summer for a few weeks to see friends/family. This past summer we wanted to do some small float trips in in Hoosier state and I will pass these suggestions on to you.
Down-town Indy canal...obviously flat-water, but fun to see the city from a kayak and get used to paddling in a controled environment. Beautiful at night with all the down-town lights. You can paddle 1 mile one way and .5 mile the other way, but spending a few hours in the kayak and 3 miles or so will do wonders for your skills. I would start here...real easy, park by the down-town fire department and carry your boat down the stairs to the canal.
Sugar Creek...Out West about 2 hours from Nap-Town is the Shade St. Park and Turkey Run St. Park. You can paddle Sugar creek almost year round and it is a beautiful float down a relaxing creek/river like area under covered bridges. This would be a great introduction to moving water with a few class I ripples and a couple of places where you would have to make a move or two. The cool thing about this float is that you can do it easily in a few hours through turkey run state park, or put in up at Shade St. Park and float for about 25 miles down to the take-out. Down the road this could be a cool overnite trip for you and some friends. I would recomend googling it and even having a conoe support for all your gear/beer.
White-River...A lot of Hoosier boaters float this right out the Indy door. As scenic as Indiana can get in a lot of places, but with boat traffic this can be a little daunting for an inexperienced paddler...work your way up to this or do it with friends through that club you were talking about.
Lake Monroe...I spent my college years paddling daily out on this beauty of a lake near Bloomington, a little bit of a drive from Nap-town, but worth it especially in the fall when the colors change. I would recomend getting a National Geographic Toppo for this area (Hoosier Natl Forrest) from a Gander Mountain or Dicks Sprting goods and checking this out. Camp along the shore and float along the lake. Great open water experience and awesome sights. I have been on canoe trips out there and ran into numerous kayakers doing self-support weekend trips...tons of hiking and even some caves to find. We once paddled up one of the tributaries of the lake into Brown County for miles and miles one night when the lake level was high. Great spot to work on your roll.
Wabash...Many Hoosier floaters seek this destination for a nice day/overnite float. Check up near Purdon't and see about floating down the river for a while.
Finallly, up in Broadripple is the best out-door rec. store in Indiana with decent people and great gear at fair prices. Call them for more info, they have a nice book section with all of these options and much more. Check out Indiana Rivers and Creeks book by this one married couple, I don't have their names but it is a good read. They will point you in fun directions.
Have fun man and be safe, also...once you get to that level there is supposed to be a top notch white-water play park near the Notre Dame campus in South Bend... I have no first-hand knowledge of it, but it comes highly recomended from mid-west whitewater paddlers...work your way up to that one!