Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries:

(1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup

(2) Euclidean: 3 torus $\mathbb{R}^3/\mathbb{Z}^3$

(3) $\mathbb{S}^2 \times \mathbb{R}$: 2 sphere times a circle $ S^2\times S^1 \cong (SO_3/SO_2) \times SO_2 $ or projective plane times a circle $ \mathbb{RP}^2 \times S^1 \cong (SO_3/O_2) \times SO_2 $

(4) Nil: Take the three dimensional real Heisenberg group, which is just the group of upper triangular $3\times 3$ real matrices with all 1s on the diagonal, and mod out by the subgroup with integer entries. This quotient is called the "Heisenberg nilmanifold." This is an iterated principal circle bundle like all compact nilmanifolds in any dimension (https://arxiv.org/abs/1805.06585).

(5) $\widetilde{\mathrm{SL}_2(\mathbb{R})}$: $ SL_2(\mathbb{R}) $ mod a Fuchsian surface group gives a unit tangent bundle (a type of circle bundle) over a hyperbolic surface.

Now I want to find examples for the other 3 geometries of a matrix group $ G $ mod some cocompact subgroup $ H $ whose quotient is a 3 manifold with the desired geometry.

(6) Sol: Could someone suggest a $ G $ and an $ H $ such that $ G/H $ is a compact 3 dimensional manifold with Sol geometry? (EDIT: this is answered here https://math.stackexchange.com/questions/4317139/bianchi-classification-of-solvable-lie-groups-and-cocompact-subgroups )

(7) $ \mathbb{H}^2 \times R $: Could someone suggest a $ G $ and an $ H $ such that $ G/H $ is a compact 3 manifold with $ \mathbb{H}^2 \times R $ geometry ?

(8) Hyperbolic $ \mathbb{H}^3 $: Could someone suggest a $ G $ and an $ H $ such that $ G/H $ is a compact hyperbolic 3 manifold?

(EDIT: As Moishe Kohan states in his answer every compact homogeneous 3 manifold is either $ G/H $ for $ G $ 3 dimensional and $ H $ discrete (which covers my examples here for all geometries except $ S^2 \times R $) or if it is not of this form then it must be a fiber bundle of the following sort: either a bundle of circles over a sphere/projective plane/torus/klein bottle or a bundle of sphere/projective plane/torus/klein bottle over a circle)

5more comments