There's nothing particularly strange about homotopy groups or homology groups having a countable infinity of generators (unless the manifold is compact, as said in the comment of @LordSharktheUnknown).

For example the ladder surface is the boundary of a regular neighborhood in $\mathbb R^3 = \mathbb R^2 \times \mathbb R$ of
$$\bigl((\mathbb R \times \{0,1\}) \cup (\mathbb Z \times [0,1])\bigr) \times \{0\}
$$
The ladder surface has countably generated $\pi_1$ and $H_1$, but neither is finitely generated. Perhaps an even simpler example is $\mathbb R^2$ minus the radius $1/3$ balls centered at the points of $\mathbb Z \times \{0\}$.

Here's a general fact along these lines. Connected smooth manifolds are usually required by definition to be paracompact (this is unnecessary for Riemann surfaces, which are *proved* to be paracompact by Rado's Theorem). One can then use a partition of unity argument to prove that there is a "good open covering" by open sets homeomorphic to balls which is locally finite and such that any finite subset of the covering intersects in either the empty set or a subset homeomorphic to a ball. By applying combination theorems (e.g. Van Kampen's theorem for $\pi_1$ and the Mayer Vietoris theorem for $H_n$, perhaps coupled with direct limit arguments) one can then conclude that $\pi_1$ and each $H_n$ have countable generating sets (any of them could, nonetheless, still be finitely generated). By applying somewhat deeper combination theorems one can also prove that the higher homotopy groups $\pi_n$ have countable generating sets.