+2 Until end of turn, Gideon, the Oathsworn becomes a 5/5 white Soldier creature that's still a planeswalker. Prevent all damage that would be dealt to him this turn.
Does "this turn" mean I can use this ability on both my turn and my opponents turn? In other words, can I increase his loyalty quickly by activating this ability during each "turn"? Or can I only do this when it's my turn?
The ability does not state any exception to the standard planeswalker rules, so you can't activate it on your opponent's turns unless something else does allow you to do so.
306.5d Each planeswalker has a number of loyalty abilities, which are activated abilities with loyalty symbols in their costs. Loyalty abilities follow special rules: A player may activate a loyalty ability of a permanent they control any time they have priority and the stack is empty during a main phase of their turn, but only if none of that permanent’s loyalty abilities have been activated that turn. See rule 606, “Loyalty Abilities.”
Notably, Gideon's two loyalty abilities can't be activated unless it's Gideon's controller's turn (and other requirements are met), regardless of what card types Gideon has (so even if Gideon isn't a planeswalker at all) (C.R. 606.3). See also this thread and this thread. Gideon's last ability doesn't change when a loyalty ability can be activated (compare Gideon with The Chain Veil).
Also, as far as the game is concerned, "this turn" and "until end of turn" mean the same thing (C.R. 514.2). Thus, the effect making Gideon a creature ends at the same time as the effect preventing damage to him.
Does "this turn" mean I can use this ability on both my turn and my opponents turn? In other words, can I increase his loyalty quickly by activating this ability during each "turn"? Or can I only do this when it's my turn?
Also, as far as the game is concerned, "this turn" and "until end of turn" mean the same thing (C.R. 514.2). Thus, the effect making Gideon a creature ends at the same time as the effect preventing damage to him.