Let $H$ be this connected component. You first need to show that this is even a subgroup. You have a continuous map $H \times H \to G$ given by $(x, y) \mapsto xy^{-1}$, and you want to show that the image is contained in $H$. What are all of the definitions involved?
- A product of two connected spaces is connected.
- The image of a connected space under a continuous map is connected.
- A point is in the connected component of $e$ if and only if there exists a connected subset of $G$ containing that point and $e$.
I think your idea for proving that $H$ is normal is a fine one. Proving all of this for the path component of $e$ can be very similar.