$A^\circ$ need not be connected even if $A$ is connected. (in $\mathbb{R}$)
I realize that a subset of $\mathbb{R}$ is connected iff it is an interval. This to me means that interior points are connected, even if they are singletons, because they are all limit points essentially.(?)
However, I have been reading some conflicting information stating that this may not always be the case. Can anyone enlighten me? Thank you.
In $\mathbb{R}$, the interior of a connected set is connected. As you remarked, this is because the only connected sets in $\mathbb{R}$ are intervals, and the interior of an interval is an interval (or empty).
In $\mathbb{R}^2$, this is not true. Consider, for example, two tangent disks. This is a connected set, while the interior has a disconnection because of a single point missing where the circles were tangent.