How do I go about finding a formula for the number of tiles I would need to add to an arbitrary square to get to the next sized square?

Get some squares or graph paper and try it. Don't look for a formula but rather watch what you do.

Here is a 3 by 3 square.

Now try a 5 by 5 square

In each case you add a row on the top (orange), a column on the right (orange) and one more tile (red). That is, for a 3 by 3 square you add

For an arbitrary square, say one that is n by n, how many tiles do you add?