liljohn,
If you want something to do a little off green laning or off roading but still behave themselves onroad, an all terrain (AT) tyre with something like a 50/50 on/off road bias is ideal. ATs are very popular for green-laners, BF goodrich being a popular brand.
If you wanna do some serious off roading in the mud (off roading pay n play sites for example), then really you should be looking at Mud Terrain (MT) tyres. These tend to have an 20/80 on/off road bias, ie they are sods on the road- very noisey, feeling like your wheel bearings have gone, shake the fillings out, and a bit scarey at anything above about 50mph....but amazing off road! (you'll be rescuing the boys n girls on AT tyres long before you get stuck yourself). I've been very happy with the Insa Turbo Dakar tyres fitting to my off road toy, but I wouldn't want to go at any speed on a motorway with 'em.
good luck. Have a chat with your local tyre centre, and/or do a bit of looking here on the internet.