Is the amount of energy different or the same when you are climbing straight up the fall line or if your traversing across it at the same % gradient?
Energy is the same either way, the same as the potential energy due to the total difference in height.
Power is the rate at which you use energy, which is why you feel it is harder to climb straight up. For a given vertical distance it takes more power to climb it in a shorter time (ie, feels harder).
The angle of your tires makes no difference with regard to energy or power expended. The orientation of your body with respect to gravity also does not change the energy needed to climb.
work: don't get confused by the terminology and equations. Work in this case is the change in potential energy.