You could have the skill call up a common event which just had the script call which varies the speed. The script call is:
$game_player.move_speed = x
where x is the speed you want. Default speed is 4 and dash is 5.
Your problem here is that once activated, it would continue for ever, so I suggest doing this with Show Choices in the common event. Something like "Do you want to fast travel?"
When 'Yes' have the script call with the higher number, when 'No' have the script call with the lower number.
As for gaining a skill after defeating a boss, I assume this is an evented battle. After the battle, use the 'Change Skills' command on the right side of the first tab in the Event Command Menu. Pick the actor who is to gain the skill, select 'Learn', select the skill.
We urge people to have one question per thread. That is partly to avoid confusion when people are not quite sure which question is being replied to, but also because in the future someone else might have the same query, but wouldn't know from the title of this thread that it also dealt with gaining skills. So in future, please keep to one question, one thread.
Thanks.