Where do I start when it comes to speed running?
I think it starts at starting up your console (y)
On a more serious note: The best thing to do for an established speedrun like Splatoon would be to watch some speedruns of it. That way you can see and learn the current strategies of the game first hand and learn the route of the game. After that I would look for a (at least mostly) up to date tutorial for running the game. These often are better for learning, and are usually explained in detail, with button presses, movement and such.
After that it's basically practice, practice, practice.