I assume that all manifolds are closed and connected (embedding in $\mathbb R^n$ or in $S^n$ is the same), but not necessarily orientable.
Concerning dimension 3, a famous theorem of Wall states that every closed 3-manifold embeds in $S^5$.
It is not possible to improve this result. A theorem of Shiomi shows that there is no closed 4-manifold which contains every possible closed 3-manifold.
The question in dimension 4 seems more complicate. Since $4=2^2$, this is one of the dimensions where real projective plane $\mathbb R\mathbb P^4$ embeds in $S^8$ and not in $S^7$. Every orientable 4-manifold embeds topologically in $S^7$ by a theorem of Fuquan Fang.
