Pride of the Road

“Max speed — [ability]” means “As long as you have max speed, this object has [ability].” If the granted ability functions in a zone other than the battlefield, the max speed ability does too.

Start your engines! isn’t a triggered ability. Increasing your speed to 1 is something that happens as a state-based action as soon as you control a permanent with the ability. Notably, this includes gaining control of a permanent with the ability that another player controls.