Although theoretically one can do canyoneering alone, in reality it's not something you can safely start doing all by yourself, especially since you don't have mountaineering or rock climbing experience. So the first thing to do is to try to locate someone more experience with whom you can go. Canyoneering can be dangerous, basically because it involves lots of rappelling. If you read Accidents in North American Mountaineering for a randomly chosen year, you will see lots and lots of rappelling accidents. Canyoneering is also difficult to do safely because a trip down a canyon is committing. Once you've done the first rappel, you generally have no choice but to complete the remaining rappels. If you get to a certain rappel and find that the anchor is unsafe, you have a serious problem. Wet canyoneering adds to the danger. For all these reasons, it would be extremely foolish for someone with your skill set to try to do this without guidance from someone who is going to be there with you.
Assuming that you're going with someone else who is more experienced, the minimum personal equipment you would need for yourself would be a climbing harness, pear-shaped locking carabiner, rappel device, helmet, cordage for making a backup Prusik, and some kind of Prusik (such as a Texas Prusik) to carry on your harness in case something goes wrong and you need to ascend the rope. The other person would presumably provide rope, rope bag, webbing for anchors, extra oval carabiners for misc purposes, and other shared equipment.
To build up a minimal set of skills, it would be a good idea to do a little rock climbing, either at a gym or outdoors. You should buy the standard textbook on mountaineering, called Freedom of the Hills, and study the chapter on rappelling extremely carefully. You also need to learn how to ascend a rope on a Prusik and how to stop a rappel with a leg wrap.
There are some additional skills that your leader might have, but that it would also be good for you to have. These including building an emergency rappel setup and building an anchor. Anchor building is an extremely technical skill. Although you may intend to go on canyoneering routes where all the anchors are supposed to be pre-built (e.g., there are bolts at every rappel station), in reality you can't assume that they will all be OK. If you get there and the anchor is damaged or appears unsafe, you need to know enough to deal with the situation.
climbingtag since no tag exists for
canyoningand I am not allowed to create it and I think
climbingis probably the closest. I find
canyoninga lot more appropriate, though.
canyoningso now it's tagged properly. And by the way, welcome to The Great Outdoors!