Basically we were given vector problems, HELP!

Concept is called vector addition. The unknown desired heading should be the vector that when added to "east wind is blowing at 65mph" yields, "260mph on a bearing of 180 degrees". Split them into NS and EW components.

 

Nothing wrong with that. Taking "east" to the right and "south" downward, draw a line to the right with length "65 mph" and, at the end of that, draw a line downward with length "260 mph". That gives you a right triangle with legs of length 65 and 260. The airspeed is the length of hypotenuse. The Pythagorean theorem gives you that. Which angle you use depends on how you want to write the course. If you want to write it as a compass reading in degrees from north, it will be 90 plus the angle in the upper left of the triangle, where you started. If you want to write it as "degrees east of south", it will be 90 degrees minus that angle. In either case you can use the tangent function to find "that angle".