I'm beginning to study some algebraic topology and I'm having some trouble working out the example of the torus. I want to calculate its Betti numbers, which are described as the "$k$-dimensional holes" of a surface. The rigorous definition is the rank of the homology groups $H_n$. So if I want to calculate $\beta_0, \beta_1, \beta_2$, then the approaches I can think of are either trying to do it from the former intuitive definition rigorously, or find a triangulation of the torus and calculate it by hand.
I can't figure either of these out. I can't tell if I'm missing something or if a triangulation of the torus is nontrivial, but the fact that the intuitive answer seems so simple suggests to me there should be a simpler way (as opposed to finding a complicated triangulation) to calculate them rigorously, either by making the intuition rigorous or by some other method. I've also read that the Poincare polynomial somehow gives the Betti numbers for $T^n$ by $(1+x)^n$; does there underly an alternative approach here which might be simpler?