Given a locally compact group $G$ and a closed subgroup $H$, there is a $G$-invariant measure on $G/H$ if, and only if, $\Delta_G|_H=\Delta_H$ (where $\Delta_G$ is the so-called modular function of $G$, which measures the difference between the left and right Haar measures). Every discrete group $H$ is unimodular (i.e. $\Delta_H$=1), so, according to the above, you'd have to check that $\Delta_G|_H$ is trivial. ~~in fact, when $H$ is a discrete subgroup of $G$, there's a $G$-invariant measure on $G/H$ if, and only if,~~ This will occur if $G$ itself is unimodular (e.g when $G$ is abelian, or compact, or a reductive group), but may occur more generally, for example with $G$ being the upper-triangular matrices in $\text{SL}(2,\mathbf{R})$ and $H$ the unipotent upper-triangular matrices in $\text{SL}(2,\mathbf{Z})$. In most cases, to induce a measure on $X=G/H$ you can simply pick a fundamental domain $F\subseteq G$, and integrate over it (according to this, this can be done at least when $G$ is $\sigma$-compact). More generally, you can normalize things so that for any integrable function $f$ on $G$
$$\int_X\left(\int_H f(gh)d\mu_H(h)\right)d\mu_{G/H}(gH)=\int_Gf(g)d\mu_G(g).$$

ADDED: I've come across a reference that could help: It's in the Encyclopaedia of Mathematical Science series, specifically "Lie group and Lie algebras II". Chapter 1 is "Discrete subgroups of locally compact groups". Here's a link to google books.