Hopefully I can go some of the way toward addressing 2 and 3 without getting carried away and getting too technical/newbie unfriendly, although I have a feeling I'm going to fail at this last part.
The answer to 3 is that yes, there are ways of making the notion of morphism of varieties (or more generally schemes) intuitively obvious (at least for me). For instance one can view a variety as a certain type of locally ringed space. A locally ringed space consists of a topological space, in this case the underlying set of points of the variety with the Zariski topology, together with a sheaf of rings which is just a way of keeping track of the regular functions on the various open subsets and how one can glue them together (plus the locally part which comes down to the fact that functions not vanishing at a point are invertible in a neighbourhood of that point). From this point of view a morphism of varieties is precisely what it has to be - a continuous map of topological spaces together with a map of sheaves which tells you what happens to regular functions when they are pulled back. This is really just a restatement of the definition you know, but from the point of view of sheaf theory there is really only one thing a morphism of sheaves can be. (One can see it "is what it has to be" via a functor of points interpretation or by gluing as Simon outlines in his answer)
As far as 2 goes two examples come to mind (although I guess this is still a sort of scheme theoretic rather than classical variety type approach). The first is that over an affine base any projective scheme comes from a graded ring via the Proj construction. Using this one can construct certain morphisms from quasi-projective varieties to projective ones from graded morphisms of graded rings in analogy with the way one gets morphisms of affine varieties from morphisms of rings.
The second comes from the adjunction between Spec and global sections - i.e. for any ring A (commutative with unit) and any variety (this works for schemes in general) X there are natural bijections
Hom_{Schemes}(X, Spec A) ~ Hom_{Rings}(A, Γ(X,O_X))
There is a special case of this which I think is informative (at least it is a useful thing to know). Suppose we are working over some fixed base field k which is algebraically closed (although this is fine over any affine base) and consider maps from X to Spec k[t] (the affine line over k). Then this bijection tells us that to give such a map is the same thing as giving a globally defined regular function on X (the global section which we send the indeterminate t to). So if one knows the globally defined regular functions on X then one knows all of the morphisms to the affine line. Conversely one can actually use this interpretation to show that the only globally defined regular functions on a projective scheme are constants (although this is probably overkill).
I hope that this is at least vaguely helpful!