Rank of SO(3) and SO(4)?

Since you're asking about the Lie groups $SO(3)$ and $SO(4)$, you're looking at the wrong definition of rank.

You don't want the rank of a group meaning the minimal number of generators; for an uncountable group, that rank is uncountable, as you suspected.

Instead you want the rank of a Lie group, and I quote from that link: "For connected compact Lie groups... the rank of the Lie group is the dimension of any one of its maximal tori."

The rank of a Lie-group is the dimension of a maximal torus. In $SO(3)$ a maximal torus is given by the rotations around just one axis, for example $SO(2)\times \{1\} < SO(3)$, and this is diffeomorphic to $S^1$, the circle. Therefore the rank of $SO(3)$ is 1 (the dimension of $S^1$).

In $SO(4)$ you have maximal tori of the form $SO(2)\times SO(2)$, so here the maximal torus really is a torus $S^1 \times S^1$, which is 2-dimensional. So the rank of $SO(4)$ is 2.

Edit: I changed maximal abelian groups to maximal tori. (Thanks for the comment)