Here is the list ordered from "required" to "recommended":

(1) The most important common topics: Cox, Little, O'Shea "Ideals, Varieties and Algorithms" Undergraduate Texts in Mathematics. Chapters 1 and 2.

(2) Less common topics: Cox, Little, O'Shea. Chapters 3 and 4.

(3) Special topics in ...

Computer Algebra: Section on primary decomposition in chapter 4 of Cox, Little, O'Shea.

Polynomial Optimization: Chapter 7 of http://math.berkeley.edu/~bernd/cbms.pdf

Numerical Algebraic Geometry: Section 2 and 3.1-3.3 of http://homepages.math.uic.edu/~jan/Articles/intro.pdf

Tropical Geometry: Chapter 1 of http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/TropicalBook.pdf

(4) In addition take a look at the following freeware; we recommend you to install these on your laptops.

(4) In addition take a look at the following freeware; we recommend you to install these on your laptops.

Singular: http://www.singular.uni-kl.de/

Bertini: http://www.nd.edu/~sommese/bertini/

Yalmip: http://users.isy.liu.se/johanl/yalmip/pmwiki.php?n=Tutorials.SumOfSquares (depends on Matlab and an SDP solver, e.g., SeDuMi http://sedumi.ie.lehigh.edu/)

Matlab: Please, check with your university if a free license is available to you.