Reference book for understanding Hilbert Series/functions

In addition to Eisenbud's book which has already been pointed out, I would like to suggest also the following references.

• Chapter 11 of Atiyah-MacDonald's "Introduction to Commutative Algebra" develops the theory of dimension for graded and local rings and its relation with the Hilbert function. The prerequisites you mention should be enough for reading the book, but for Chapter 11 you may need notions developed in the previous chapters such as Noetherian rings and the length of a module.

• Personally, I like a lot the approach of Bruns-Herzog's "Cohen-Macaulay Rings". Chapter 4 is dedicated to the Hilbert function. Although this book is more advanced and requires in several places a certain familiarity with homological algebra, I think that Chapter 4 is more accessible. So you may try to give a look also there.

• Since you mentioned Gröbner bases, I recommend also Kreuzer-Robbiano's "Computational Commutative Algebra 2". They follow a more computational approach (as the title suggests) and they dedicate a very dense Chapter (the fifth) to Hilbert functions of graded and multigraded modules. The introduction of that chapter is nice and gives some simple examples of the geometric importance of the Hilbert function.