I use Macaulay2 for standard computations in commutative algebra/algebraic geometry, like Gröbner bases, graded free resolutions, Tor/Ext groups etc. There are also a lot of add-on packages you can import for working with say, intersection theory, toric varieties and convex geometry. M2 is easy to learn and runs quite smoothly, especially using the interface in emacs.
In addition to the M2-documentation, which is very complete, there is also a really nice book, Computations in algebraic geometry with Macaulay 2 available freely on-line.
