Some time ago, a graduate student interested in commutative algebra and algebraic geometry asked me for input on what to read and how to choose a thesis advisor. For what they are worth, here are some thoughts on the subject.

This is a Springer UTM, and is a nice introduction to the area; I taught an undergraduate class out of it with great success. It is full of examples and an easy, fun read.

A very approachable introduction to the important ideas of commutative algebra , and a good lead in to the book of Eisenbud (see below).

Probably the most comprehensive introduction to curves, it looks at the subject from all angles. Miranda notes in the introduction that the book grew out of a course he has taught 5 times; this shows in the smooth presentation.

A complex analytic introduction to algebraic curves, and one of my favorite elementary books. You need only a class in one complex variable (or read, say, Churchill and Brown); it is amazing how far one can go with elementary tools.

This book is a logical follow-up to Cox-Little-O'Shea, and is also very example driven. Lots of good exercises.

Notes from an undergraduate class I taught at Harvard; basics of commutative algebra and Grobner bases, plus a quick intro to homological algebra (Ext and Tor) and a bit of sheaf cohomology.

For the student interested in connections to discrete geometry and combinatorics, the following two texts are a good start:

This book explores the beautiful connections between commutative algebra and combinatorics. Also, gives a nice introduction to homological algebra, with many concrete examples of things like free resolutions, as well as your favorite derived functors (Ext, Tor and local cohomology).

This book pairs well with Stanley's book above; it is another book with lots of examples and an informal, easy style. Great book for a grad run seminar in the summer (I did this a few years back), and useful if you will be doing anything with toric varieties.

OK, so now you've covered all the above, and need some more background/reference books. Here is the advanced reading list:

It is really important that you and your advisor get along well; grad school is difficult enough without having a chilly (or worse, hostile) relationship with your advisor. The best time to figure this out is BEFORE you ask someone if they'll work with you. One way to determine this is by taking a course with the person, and seeing how you interact when you go to office hours to ask questions. It goes without saying that you should make sure you do a good job in the class, because if you do poorly, it is likely that the potential advisor will not be interested in taking you on as a student.

It is important that your advisor be respected as a scholar. You will quickly find that one of the first questions mathematicians ask a graduate student is: "and who is your advisor?". You may also want to find out how many theses the professor has directed, and how many students have dropped out (this should be done tactfully, of course, by talking to former students or the director of graduate studies).

Most professors supervising dissertations have a set of problems that they think might make an appropriate thesis; you should ask about this, 'cause getting a bad problem can cause you to waste years of your life.