Greenham common would be the longer but better route. From the common you can cross the Basingstoke road and head towards Adbury. From there you can cut across Newtown Common passed Heather world Stud and then head towards the Wayfarers up Ladle or Sydmonton Hill (different names for the same place) From there you can ride to Combe Gibbet via Highclere and on towards Hungerford.
The other way would be to take the canal towpath through Newbury and towards Kintbury. Just before Kintbury you can turn off at Sheppards bridge and go up Irish Hill and head towards Inkpen. Cross Inkpen Common and head towards Combe Gibbet and the Wayfarers via West Woodhay and go up Adders Copse for a bit of a climb.
you should be able to find those place names on a local 1:50000 map and there’s a whole network of tracks a lanes round there.