I would get a 1:25000 OS map of the area, Loads of Bridle ways on either side of the A22, from the Burford bridge Carpark you can head out via the roman road up onto White Hill and Mickleham Downs and head out to Headley Heath Food stall there most days, the tracks around High Ashurst outdoor center will lead you back to the Box-hill village.
On the other side of the A22 you have Ranmore common, best to use the Carpark up on the Common, there are plenty of tracks then you can head out to Newlands Corner or Guildford. Or you can head down to the Leith hill area and plenty of tracks around there, Cafe at tower at weekends.