This gem is perhaps the best place to get introduced to the fundamentals of algebraic curves and projective curves. The exercises illuminate the concepts throughout the text. The pictures are chosen wisely, too. Where else can you get Riemann-Roch in less than 110pgs of well written mathematics? Not to mention a neat intro. to intersection multiplicities (Fulton won the Steele prize for his later book, 'Intersection Theory'.)

Prereqs: def. of a ring (really! look at 1.1 in book) and patience.

Although schemes are not mentioned (why should they be in an intro book?), I Highly recommend this book as a first step in learning algebraic geometry.

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

My only complaint: change "Max Noether" to "Emmy Noether". :)