There is an epimorphism $\mathrm{SU}(2) \times \mathrm{SU}(2) \to \mathrm{SO}(4)$ with the kernel $\langle(−I, −I)\rangle$. Since $\mathrm{SU}(2)$ is isomorphic to the unit quaternions, the epimorphism is given by $(u,v)\mapsto R_{u,v}$ where $R_{u,v}$ is the rotation of $\mathbb{R}^4$ given by $R_{u,v}(q)=v^{-1}qu$ for any quaternion $q$.

And $\mathrm{SU}(2)$ maps onto $\mathrm{SO}(3)$ with kernel $\langle -I\rangle$; again use quaternions.

As you say, you know the closed subgroups of $\mathrm{SO}(3)$, and so this gives the closed subgroups of $\mathrm{SU}(2)$ and so those of $\mathrm{SU}(2)\times \mathrm{SU}(2)$ and finally those of $\mathrm{SO}(4)$.

More generally, all simple subgroups of real Lie groups are known, as described here:

Karpelevič, F. I. The simple subalgebras of the real Lie algebras. Trudy Moskov. Mat. Obšč. 4 (1955), 3–112.

Karpelevič, F. I. Classification of the simple subalgebras of the real forms of classical algebras. Doklady Akad. Nauk SSSR (N.S.) 93, (1953). 613–616.

Karpelevič, F. I. Classification of the simple subgroups of the real forms of the group of complex unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 85, (1952). 1205–1208.

There is an epimorphism $\mathrm{SU}(2) \times \mathrm{SU}(2) \to \mathrm{SO}(4)$ with the kernel $\langle(−I, −I)\rangle$. Since $\mathrm{SU}(2)$ is isomorphic to the unit quaternions, the epimorphism is given by $(u,v)\mapsto R_{u,v}$ where $R_{u,v}$ is the rotation of $\mathbb{R}^4$ given by $R_{u,v}(q)=v^{-1}qu$ for any quaternion $q$.

And $\mathrm{SU}(2)$ maps onto $\mathrm{SO}(3)$ with kernel $\langle -I\rangle$; again use quaternions.

As you say, you know the closed subgroups of $\mathrm{SO}(3)$, and so this gives the closed subgroups of $\mathrm{SU}(2)$ and so those of $\mathrm{SU}(2)\times \mathrm{SU}(2)$ (via Goursat's Lemma) and finally those of $\mathrm{SO}(4)$.

More generally, as it relates to semisimple subgroups, all simple subgroups of real Lie groups are known, as described here:

Karpelevič, F. I. The simple subalgebras of the real Lie algebras. Trudy Moskov. Mat. Obšč. 4 (1955), 3–112.

Karpelevič, F. I. Classification of the simple subalgebras of the real forms of classical algebras. Doklady Akad. Nauk SSSR (N.S.) 93, (1953). 613–616.

Karpelevič, F. I. Classification of the simple subgroups of the real forms of the group of complex unimodular matrices. Doklady Akad. Nauk SSSR (N.S.) 85, (1952). 1205–1208.