LambofGod--you got a few partial answers.
And while 'keith smith's' answer may be technically correct it's pretty hard to visualize (Apologies Keith, but at this point images work better than equations).
First, you need to understand the concept of vectors. I'm going to assume you have a handle on this concept. If not, then do some research on it and if you still have questions then follow up here.
Second, Mark1234 is correct when he states that the lift produced by the wing in a climb is slightly less than in level flight due to the VERTICAL COMPONENT OF THRUST.
That is to say, when you point the nose up, thrust also has an upward vector.
In fact, airplanes climb due to EXCESS THRUST, not lift.
To answer your question about induced drag--another way to think of induced drag is to describe it as "the rearward component of lift" generated by the wings.
In a very basic way, an airfoil needs a positive angle of attack in order to produce lift (this is not the Bernoulli type of lift, but more Newtonian [for every action there is an equal and opposite reaction]--it's just that sometimes it's more convenient to describe the lifting force using Bernoulli; and sometimes more convenient to use Newton--both are valid, depending on the situation).
As flight students we're taught that LIFT (with a capital L) directly opposes Gravity (with a capital G). But that's not the entire story. In reality, it's only the VERTICAL COMPONENT of Lift that opposes Gravity. TOTAL LIFT is actually pointed slightly AFT, due to the slight angle of attack. The REARWARD COMPONENT of Lift is what we call INDUCED DRAG. It's Lift that's working to the rear instead of strictly working towards the vertical.
So, as you increase your angle of attack, you gradually point that big old TOTAL LIFT more and more to the rear--or aftwards and your rearward component of Lift grows greater and greater, increasing your induced drag.
If you draw a "Drag Curve" (Total Drag vs. IAS) it would look like a capital "U". Why? Because in high speed flight your drag is high thanks to Parasite Drag (Increases to the square of your IAS). But at slow speeds your drag is high due to Induced Drag.
That's why your best economy airspeeds are somewhat right in the middle where it's the best compromise of low Induced Drag and low Parasite Drag.
I hope all of this makes sense. It would be much easier with a chalkboard, but if you have some good textbooks maybe you can find the illustrations I'm describing.
Good luck.