You're best off with trail/fell shoes as they have better grip and it's good to have shoes that release water quickly. Me and the Other Half have Innov 8 (£80 - £100) and they've been good. Only thing is like you I'm an overpronator and I have yet to find any grippy shoes that are as supportive as my running shoes. My sports therapist has recommend seeing a podiatrist to get inserts.
Kit - best to wear clothes that dry quick and don't fall down under weight of water. I think sleeves are essential after rubbing my underarms raw going over A-Frames. A lot of people wear as little as possible - vests and shorts. Also gloves can be helpful to give you grip when you hauling yourself over the obstacles.