I live in Churt too.
Alice Holt forest is reachable almost exclusively off road by going down a bridleway near the Frensham Pond Hotel, follow it to the end and you come out in Frensham village on the bridge over the river, turn left and ride for 200 yrds on the road then take next bridleway on the right and follow this to the end (Bluebell Pub in Rowledge), turn right at the pub and ride for another 500 yrds on the road and then take any number of routes off to the left into Alice Holt....Alice Holt is ok, the main routes are fire roads type stuff, head off into the trees and there is some decent singletrack that can be found....i once had an excellent ride there when i crossed the road near the entrance and went into the woods opposite, much better.
Whitmore Vale is good, its bigger than teamhurtmore thinks...he just needs more time to explore it!....the trick here is that it spans a couple of places and requires a road crossing at one point and then there's loads more to ride.
Ludshott common....doesnt get much credit, decent place...nowhere near as sandy as other places in this area, stunning views, the dirt jump kids have built some handy drops/jumps etc in the trees on the outskirts of here too but i found them once and havent managed to again!....connects nicely to Waggoners Wells too for a longer ride.
Frensham Common (big and little ponds)....ok-ish...sandy, flat...more of a relaxing evening ride thing.
Thursley Common...sandy in places but hard enough on most of it to ride on, spent several hours over there one day in the summer, bigger than you think, some short sharp hills/descents....joins nicely onto Hankley common.
Devils Punchbowl....pretty good but the middle is boggy and often blocked by logging, mainly ride around the edge, nice views, shouldnt really ride there as there are signs saying no bikes but the walkers dont seem to mind when i've been there...on the other side of the road is....
...Gibbet Hill, great place. Used in the latest Whyte Bikes video, went there a few weeks ago...bugger to find, had to hot-foot it across the old A3 and avoid the Hindhead Tunnel workers to get to it, did see a tarmac road leading to where i emerged from the bushes and asked some walkers where it went....they said to the Devils Punchbowl so the two must link up somewhere?
Bourne Woods, where they filmed the battle scenes at the start of Gladiator. Few miles from Churt on the road or just drive (its about 3 mins in the car)...enjoy this area too but dont seem to go there very often despite always enjoying it?!....quite big and easy to while away a few hours.
Queen Elizabeth Country Park....ok not technically in or around Churt but only 20 mins down the A3 and well worth a visit, check out the thread on here by the trail building group, they are doing some good work.
Ceasars Camp....Farnham/Fleet/Aldershot....rode there once from Churt, involved a bitch of a road climb up Castle Street in Farnham so i was too knackered when i got there to enjoy it properly...i'll drive next time!....nice single track, drops/jumps etc...was used recently for the Enduro type event that got a bit of a slating on here but dont let that put you off.
Swinley is also no more than 30mins away in the Frimley/Camberley direction but i've never been so couldnt tell you much about it.