You also have to realize that different people will just learn differently, and you'll have to adjust to that.
For me, I wanted a bomber roll before I was getting out of a pond and on to a river. I wanted to be able to stay in my boat, and that gave me confidence through eddy lines, ferrying, and rapids.
For my SO, she has been down class II / III- a dozen or so times and while she has her roll in a pond, she immediately pulls if she gets flipped. However, she's learning a lot about the mechanics of paddling, eddies, etc. We just need to add her roll into the formula and she'll hit the next plateau.
I have a friend who has just bullied his way into boating Class IV water, and has swam more times than I can count. For better or worse, that was his method.
Obviously the advice given in this thread is best - find a boat that fits, get some instruction, learn a roll, and then get on the river. It's the safest and most practical.
But for some people, it just takes them a long time to learn a roll (others get it in the first day), and you can still get comfortable being in a boat and using a paddle on tame class II stretches (if you have them).