I want to do the thing with the two sticks. What is that called?
That is called skiing! Skis are two individual planks attached to each foot and the skier moves down the slope facing forward. Skiing uses similar movements to roller blades, ice skates and water skis and is generally considered to be the easier downhill winter sport to pick up and enjoy.
I want to do the thing like skateboarding or surfing. What is that called?
Snowboarding, dude! A Snowboard is attached to both feet and the rider travels downhill standing sideways. Compared to skiing, there is a steeper learning curve but skateboarders, surfers and wakeboarders will have a leg-up on developing the necessary balance and muscle-memory.