Is connectedness $\implies $ local connectedness?

Neither connected nor locally connected implies the other, nor do their negations. Four examples:

$\Bbb R$ is connected and locally connected.

$[1,2] \cup [3,4]$ is locally connected but not connected

The topologist's sine curve is connected but not locally connected.

$\Bbb Q$ is neither

The Topologist's Curve is connected but not locally connected. Let $I_0, I_1\subseteq \mathbb{R}$ two disjoint intervals, $I_0\neq\emptyset\neq I_1$. Then the union $I_0\cup I_1$ is locally connected, but not connected.

Many other answers have mentioned the topologist's sine curve as a counterexample. You may also be interested in the infinite broom, since the proof that it isn't locally connected is a bit easier.