To get a good idea of what you will be able to do with Macaulay2, see the package BeginningMacaulay2.
Macaulay2 is available from our web page http://www.math.uiuc.edu/Macaulay2/. There you will find online documentation, the source code, and precompiled versions for MacOSX, Linux, Microsoft Windows, and for various unix boxes.
After you or your system administrator has installed Macaulay2, use the information here to set up the emacs interface (the recommended way to run Macaulay2), bookmark the html documentation, and to try to run Macaulay2.