It's obvious that the sphere is an absolutely symmetric surface. It remains the same for any reflection over the planes which include its diameter and also for any rotation of arbitrary angle $\theta$ cw or ccw.
Does it follow from this that the symmetry group of the sphere in $\mathbb{R^3}$ is one of infinite order? And is this observation enough if it's true, or is there a way to categorize all this infinite symmetries to write them down? Are there more than the ones I mentioned?
The symmetry group of the sphere, including reflections, is known as the orthogonal group $O(3)$. The symmetry group with rotations only is the special orthogonal group $SO(3)$. Both have infinite order, which does indeed follow from your observations that rotations by any real angle and reflections about any plane are part of the symmetry group. The rotations form $SO(3)$, and the rotations and reflections form $O(3)$, without any missing symmetries.