Move position in smooth gradient I am decrementing/incrementing position x until it reaches y. However, this creates a rectangular-like path. I want position x to smoothly curve towards y (diagonally) on a 2D plane.

What kind of maths would I need to achieve this? 
Example image attached.
The math you need is linear interpolation. I think of a player sprite which moves from position X to position Y. Pseudocode:

playerSprite.x = Y.x + t * (X.x - Y.x)
playerSprite.y = Y.y + t * (X.y - Y.y)


where t is a value between 0.0 and 1.0 (floating point). Instead of incrementing the position components of Y you increment t till it reaches 1. When it reaches 1 the playerSprite reached point X. If t is 0, then the playerSprite is at position Y.