Your problem was not winter riding, it was lack of lubrication. Lubricant will repel water and keep controls and cables free.
Before you replace the cables, make a sketch showing the current cable routing.
Actual cable installation is simple enough. If the cable end is adjustable with a clamp then set any screw adjusters close to thier shortest length and clamp the cable end with a small amount of free play in the control lever.
Be especially careful that you do not kink the cables and remember that the cable ends need lube as much as the cable itself.