If $G$ is linear topological group, then a model for $EG$ may be taken to be an infinite Stiefel manifold. More precisely, if there is an faithful representation of $G$ into $GL_n(\mathbb{C})$, that gives a free action of $G$ on the space $V_n$ of $n$-frames in $\mathbb{C}^\infty$. $V_n$ is contractible, so $BG$ may be taken to be the quotient $BG = V_n / G$. If it's helpful to think of it this way, this is a fibre bundle over $BGL_n(\mathbb{C}) = G_n(\mathbb{C}^\infty)$, with fibre $GL_n(\mathbb{C}) / G$.
Another favorite example comes from spaces of embeddings: if $M$ is a compact manifold without boundary, then it is a consequence of Whitney's embedding theorem that $Emb(M, \mathbb{R}^\infty)$ is contractible. The group $G=Diff(M)$ of diffeomorphisms of $M$ acts freely on $Emb(M,\mathbb{R}^\infty)$ by precomposing an embedding with a diffeomorphism. Therefore a model for $BG$ is the quotient $Emb(M, \mathbb{R}^\infty) / Diff(M)$, which is often thought of as the space of subspaces of $\mathbb{R}^\infty$ diffeomorphic to $M$. One can combine this idea with the previous idea for subgroups of diffeomorphism groups.
Lastly, if there is a homomorphism $G \to H$ which is a homotopy equivalence, then of course there is a homotopy equivalence $BG \to BH$. So in the previous examples, one can for instance replace $GL_n(\mathbb{C})$ with $U(n)$, and $Diff^{+}(\Sigma)$ with the mapping class group $\Gamma(\Sigma) = \pi_0(Diff^{+}(\Sigma))$ for closed surfaces $\Sigma$.