Woods to the right coming out of leamington; there's an opening just before them on the right, small drop down then follow the path around the edge of the field over a little bridge then take a left into the woods. Not very big but a nice way to start a ride.
Wappenbury woods (the other side of Weston) you can get into either from ryton pools (gate pretty much the opposite corner to the carpark), or coming from Weston go up the road and after the short climb there's a farm on the right, on the opposite corner go through the gate down over two fields and into the woods. You can piece them together in one ride coming up through that farm!
Used to do these then out across to coventry and into binley woods then back again through ryton!