Lectures are held on Mondays, Wednesdays, and Fridays at 1.30pm in SH161. The course text is Cox, Little, and O’Shea, Ideals, Varieties, and Algorithms (Third Edition), Undergraduate Texts in Mathematics, Springer-Verlag, 2007.
Bernd Sturmfels has made his CBMS lecture notes "Solving Systems of Polynomial Equations" available. You might enjoy reading the November 2005 AMS article "What is ... a Gröbner basis?", also by Sturmfels.
Via your university account, you have access to Maple. There is an excellent tutorial by C. Hillar which should help get you started. Appendix C pp. 520-522 of the textbook contain details on how to perform Gröbner bases calculations, as does this page. Alex Cowperthwaite's Maple code to perform the division algorithm and test Buchberger's Criterion is available here.
Here are links to some free commutative algebra packages which you may wish to install on your own computer. I suggest you try AXIOM first - Appendix C of the textbook contains useful notes on how to use it. I have managed to install all the packages on Mac OS X, and can help anybody who is having problems.