I'm aware of the idea of a CW complex and how to use the van Kampen, I'm struggling with the concept however of computing the fundamental group of the 1-skeleton of spaces. For example, let us consider the simple spaces of the torus $\mathbb{T}$ and the Klein bottle $\mathbb{K}$. The CW structures of these spaces consists of one 0-cell, two 1-cells and one 2-cell.
Hence we can determine the fundamental groups of these spaces simply by looking at the 1-skeleton, consisting of one 0-cell, two 1-cells and considering the attaching maps of the 2-cell $\varphi : S^1 \longrightarrow X$.
I'm unsure of what this attaching map is and how to proceed with the computation of the fundamental group.
The generators are given by the 1-cells, the relations by the 2-cells. In the torus, you can attach a 2-cell to a square with opposite sides identified. The boundary of the 2-cell traces the path $aba^{-1}b^{-1}$. So this is the relation imposed among the generators $a$ and $b$.
The Klein bottle is similar.