Definitely off road shoes. But try and get something thats inbetween, that is good for roads too. My Off-roaders are. Got them from the Up & Running shop in Horsham, which is an excellent shop, but the Tortoise & Hare, which is on that small estate on the road from Horsham to Guildford is good for stuff.
Off-roaders are good, i took mine over Box Hill, Cissbury Ring, and they worked out well. Need to be well padded, but sometimes, even with all the grip, you can still go arse over elbow. Just pick your next step carefully.
Never realised Cranleigh had a race. Just down the road too.