  We introduce a new approach for computing the monodromy of the Hitchin map
and use this to completely determine the monodromy for the moduli spaces of
$L$-twisted $G$-Higgs bundles, for the groups $G = GL(2,\mathbb{C})$,
$SL(2,\mathbb{C})$ and $PSL(2,\mathbb{C})$. We also determine the twisted Chern
class of the regular locus, which obstructs the existence of a section of the
moduli space of $L$-twisted Higgs bundles of rank $2$ and degree $deg(L)+1$. By
counting orbits of the monodromy action with $\mathbb{Z}_2$-coefficients, we
obtain in a unified manner the number of components of the character varieties
for the real groups $G = GL(2,\mathbb{R})$, $SL(2,\mathbb{R})$,
$PGL(2,\mathbb{R})$, $PSL(2,\mathbb{R})$, as well as the number of components
of the $Sp(4,\mathbb{R})$-character variety with maximal Toledo invariant. We
also use our results for $GL(2,\mathbb{R})$ to compute the monodromy of the
$SO(2,2)$ Hitchin map and determine the components of the $SO(2,2)$ character
variety.
