connected $\Rightarrow$ path connected?

This is true for any locally path-connected space (this is the crucial property to use in the proof), in particular any manifold. Slightly more generally, for locally path-connected spaces, components and path components coincide.

Like Qiaochu Yuan said, any connected locally path-connected space is path-connected. This is because local path-connectedness implies that the path-connected components are open (this is essentially by definition: every point admits a path-connected neighbourhood, and hence is an interior point of its path-connected component), and therefore are also closed (since the complement of a component is the union of the other component, hence a union of open sets). Therefore, by connectedness, there is only one path-connected component and it is everything.