As already said in the comments, every group acts cocompactly on a single point, which is hyperbolic. But of course, this is not an satisfying example. Another source of not so interesting actions comes from combinatorial horoballs. If $G$ is a finitely generated group, let it act on one its Cayley graph $X$ (associated to a finite generating set). The combinatorial horoball $\mathcal{H}(X)$ over $X$ is the graph obtained from $V(X) \times \mathbb{N}$ where two vertices $(u,n)$ and $(v,m)$ are connected by an edge if
- $u=v$ and $m=n+1$;
- $n=m$ and $d_X(u,v) \leq 2^n$.
Then $\mathcal{H}(X)$ is a locally finite hyperbolic graph on which $G$ acts with unbounded orbits (if $G$ is infinite). Thus, every finitely generated group acts properly on a hyperbolic space. But the action is not cobounded.
Admitting a cobounded action on an unbounded hyperbolic space is rather common. Groups that do not admit such actions are investigated in arxiv:2212.14292, under the name Property (NL). Constructing examples of such groups is not so obvious.
For cocompact actions, the picture is less clear (at least for me). Of course, groups with infinite hyperbolic groups are examples, and they may be far from being hyperbolic. This includes finitely generated groups with infinite abelianisations. For an explicit family of examples, consider for instance right-angled Artin groups. Namely, given a graph $\Gamma$, define
$$A(\Gamma):= \langle u \in V(\Gamma) \mid [u,v]=1, \ \{u,v\} \in E(\Gamma) \rangle.$$
If $\Gamma$ is not a complete graph, i.e. there are two non-adjacent vertices $u,v \in V(\Gamma)$, then $A(\Gamma)$ retracts naturally onto $\langle u,v \rangle \simeq \mathbb{F}_2$. On the other hand, $A(\Gamma)$ is hyperbolic if and only if $\Gamma$ has no edge.
Again, such examples are not entirely satisfying, because the actions have huge kernels. As mentioned by ADL, amalgamated products and HNN extensions (or more generally, fundamental groups of finite graphs of groups) are examples of groups acting cocompactly on trees, but they may be far different from hyperbolic groups (e.g. Baumslag-Solitar groups). In the same vein, fundamental groups of complexes of groups can be considered. Examples I like are Higman groups: for every $n \geq 4$,
$$H_n:= \langle a_i, \ i \in \mathbb{Z}/n \mathbb{Z} \mid a_ia_{i+1}a_i^{-1}=a_{i+1}^2 \rangle.$$
These groups can be described as simple polygons of groups (with vertex-groups isomorphic to $BS(1,2)$, infinite cyclic edge-groups, and a trivial face-group). The complex of groups is developpable, so $H_n$ naturally acts on a two-dimensional polygonal complex. For $n \geq 5$, this complex is hyperbolic.
Conclusion of the story: many groups act cocompactly on unbounded hyperbolic spaces, and even more when cocompactly is replaced with coboundedly*.
(*It would be interesting to have explicit examples of finitely generated groups acting coboundedly on unbounded hyperbolic spaces but not admiting a cocompact action. It probably exists, but I do not have a simple example in mind right now.)
