A king is placed on the chessboard starting where the blue circle is (1,1). He can move to the right, up or diagonally.
The ending point is marked yellow(8,8).
How many ways are there for the king to get from (1,1) to (8,8) if (3,4) has to be included in the path?
I understand what happens when I'm limited to moving up and right, but I don't know where to start here
Assuming that your "diagonally" refers to "diagonal up-right", it's fairly easy to make the route options calculation in Excel:
Equally you could do this as two separate problems, one to cross a $3\times 4$ space ($25$ options) and one to cross a $5\times 6$ space ($681$ options) and then multiply.
Taking this as two route-option problems "glued together", you can see patterns reminiscent of Pascal's Triangle in the bottom left of the diagram.
These are the Delannoy numbers forming the Delannoy array or "tribonacci triangle". Essentially you will always have some summation to do to calculate them.