Find the fundamental group and the homology groups of the 3-dimensional torus, T3 = S1 X S1 X S1.

The fundamental group preserves products, so it's Z X Z X Z right? The first homology group would be the abelianization of this group... but how do you compute higher homology groups of this surface? I came across a method called the Mayor-Vietoris sequence, but I'm not sure how to apply it here.