If you want to ride from the centre of Leeds, there isnt too much that can be done locally. Most of the good stuff is north of Bradford/Otley so you'll need the car.
Best bet is to ride up the Otley road (race the cars) up to Golden Acre park, and then you can follow the Meanwood trail back into town.
The trail is ok, but if the weather stays as it is, its probably going to be too muddy.
As for climbing there are 3 good places in Leeds;
The Leeds Wall – Climbing and bouldering wall 2 miles from the centre.
City Block – just bouldering 1 mile from the centre
Depot – just bouldering 5 miles from the centre. worth the extra distance
Gladly show you around.