Before Hartshorne's book there was Mumford's Red Book of Varieties. I think it is a great introductory textbook to modern algebraic geometry (scheme theory).
I found that Mumford is quite good at motivating new concepts; in particular I really enjoy his development of nonsingularity and the sheaf of differentials. I think another great aspect about this book is that it emphasizes how to define things intrinsically (i.e. without reference to a closed or open immersion into affine space) but also explains how to make local arguments (i.e. using immersion into affine space). A classic example of the above:
(non intrinsic tangent space): Say X is a variety and p is a point of X. Choose an affine neighborhood so that p corresponds to the origin. Then this affine neighborhood is spec k[x1, ..., xn]/I for some ideal. Let I' be all the linear terms of I (i.e. if I = (x,y^2), then I' = (x)). Then the tangent space at p is spec k[x1,...,xn]/I'.
(intrinsic tangent space): Let m be the maximal ideal of the local ring of the structure sheaf at p, then the tangent space is the dual of the vector space m/m^2.
Taking spec of the symmetric algebra of the latter gives you the former.
Some drawbacks. This book doesn't cover nearly as much as Hartshorne's book. It doesn't have that many exercises. The notation is slightly different; integral finite type schemes are called pre-varieties and you can remove the `pre' if it's also separated. Nevertheless I think its a great compliment to reading Hartshorne.