Something is ahead of you if it is in front, or perhaps in the future. For example, "you are getting too far ahead of me" or "my schedule is running ahead of time".
Two things are beside each other if they are next to each other, facing the same direction. The two women on the bench are sitting beside each other, or side by side.
Corners are where two or more surfaces or lines meet, usually at right angles. A square has four corners, a cube has six, and a circle has none. The frogs are hiding in the corner of their tank.