In the closed case, the easiest way to do this is to construct surface bundles over a circle. For example, if I let my surface $S$ be a hyperbolic surface (say a 2-torus), then the fundamental group of my surface bundle will be an HNN extension of $\pi_1(S)\cong\langle a,b,c,d\ |\ [a,b][c,d]\rangle$. If I let the generator of $\pi_1(S^1)$ act periodically, I'll get a Seifert-fibred space, and otherwise, I'll get a hyperbolic space. Hyperbolic space admits an $\mathbb{H}^3$ geometry. The Seifert-fibred space admits either an $\mathbb{H}^2\times\mathbb{R}$ or $\widetilde{SL_2(\mathbb{R})}$ geometry, depending on whether it is (virtually) a trivial circle bundle over some surface or not. [That is, if it's Euler number is zero or not.]
Now to get the $\mathbb{H}^3$ geometry, simply pick a non-periodic (outer) automorphism of $\pi_1(S)$. So you can get a 3-manifold $M$ with the following fundamental group, for example: $ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=1, a^x=ab, b^x=b, c^x=c^b, d^x=d^b\rangle.$
To get the $\mathbb{H}^2\times\mathbb{R}$ geometry, just take the product $S\times S^1$; this gives a 3-manifold $M$ with fundamental group $ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=1, a^x=a, b^x=b, c^x=c, d^x=d\rangle.$
For the $\widetilde{SL_2(\mathbb{R})}$ geometry, I don't know how to give a nice HNN presentation. But since it is really just like the $\mathbb{H}^2\times\mathbb{R}$ case, but with non-zero Euler number, I can simply add a singular fibre above to get a 3-manifold $M$ with presentation $ \pi_1(M)\cong\langle a,b,c,d,x\ |\ [a,b][c,d]=x^2, a^x=a, b^x=b, c^x=c, d^x=d\rangle.$