Here is a way to think about both the interior of a set and the closure of a set.
The interior of a set is the ‘largest’ open subset of the set.
The closure of a set is the ‘smallest closed set containing the set as a subset.
Here is another suggestion. If you can find the small book Elementary Theory of Metric Spaces by Robert B Reisel, then that would be a great introduction for you. It is written as a self-study or Moore-style class.