There are a pair of double covers $\text{SU}(2) \times \text{SU}(2) \to \text{SO}(4) \to \text{SO}(3) \times \text{SO}(3)$, and the first resp. the second more or less reduces the classification of finite subgroups of $\text{SO}(4)$ to the classification of finite subgroups of $\text{SU}(2)$ resp. $\text{SO}(3)$ (these classifications are in turn more or less equivalent and well-known) using Goursat's lemma. This classification can be found more explicitly in Conway and Smith's On Quaternions and Octonions (Section 4.3).
There is also a double cover $\text{Sp}(2) \to \text{SO}(5)$ which reduces the classification of finite subgroups of $\text{SO}(5)$ to the classification of finite subgroups of $\text{Sp}(2)$. This is probably easier but I don't know if it's well-known (in any case I don't know it).