Ian Morrison wrote up some nice lectures in the book Lectures on Riemann surfaces,World Scientific publishers, Proceedings of the college of Riemann surfaces in 1987, at the ICTM in Trieste. They were intended as an informal introduction to the two detailed treatments mentioned below by Mumford (l'Enseignement) and Gieseker (Tata).
There is a nice treatment of the chow coordinates of a projective variety in chapter 1 of the book Basic algebraic geometry by Shafarevich. This is very elementary and readable.
There is a good discussion of the existence of the Hilbert scheme in Mumford's book Lectures on curves on an algebraic surface, Annals of math studies #59. Sophisticated, but we were able to use it in a seminar long ago, and got some good insight from it.
Mumford (notes by Morrison) first wrote up the case of stable curves in Stability of projective Varieties, in l'Enseignement mathematique, 1977, based on an idea of Gieseker.
Then Gieseker himself presented his version at the Tata Institute in Bombay (TIFR), and wrote it up in their series of lectures on mathematics and physics, #69, 1982.
The original presentation of the concept of stable curves, due to Alan Mayer and David Mumford, is in talks by Mayer and Mumford at the Woods Hole conference 1964, available on James Milne's web site at Michigan, or that of roy smith (mathwonk) at University of Georgia.
As I recall, even the detailed works by Mumford, (GIT, Enseignement), always include some introductory examples and motivation that anyone can read, so one should not shy away from the actual definitive works completely. In regard to the fine recommendations above, Mukai's is actually a textbook as requested, and not a monograph like most of my recommendations here, but that of course makes it longer.
For beginners, I would observe that the Chow approach is to characterize a projective variety by all lines meeting it, thus getting a subset of the Grassmannian of lines, while the Hilbert approach is to describe a variety by the set of all hypersurfaces of fixed large degree containing it, thus getting a subspace of the vector space of those polynomials, another Grassmannaian. Then to characterize abstract varieties, one first chooses some natural projective embedding, say by a multiple of the canonical class, then considers the corresponding Hilbert or Chow scheme, and tries to collapse together all different embeddings of the same variety, in GIT by taking a quotient by a group action. This then leads to singularities at orbits which are smaller than usual, i.e. at points with non trivial isotropy coming from automorphisms of the variety. These isotropy groups are included in the data of a moduli "stack", but were always considered informative even earlier.