Can you activate a planeswalker ability the turn it enters the battlefield?

Planeswalkers do not have summoning sickness. The rule that prevents a creature from activating tap abilities or attacking (CR 302.6) applies only to creatures, not to planeswalkers. A planeswalker can use a loyalty ability the very turn it enters the battlefield.

You may activate a loyalty ability of a planeswalker you control any time you have priority during your main phase and the stack is empty (CR 606.3b). The only key restriction is that you may activate no more than one loyalty ability of any single planeswalker per turn, regardless of whose turn it is (CR 606.3).

Loyalty abilities are activated abilities with a loyalty cost (written as [+N], [0], or [−N]). Paying the cost adds or removes the stated number of loyalty counters. You must have enough loyalty counters to pay a minus cost, and the planeswalker must survive to 0 or above after the cost is paid (CR 606.5b).

Example: You cast Gideon, Ally of Zendikar (enters with 4 loyalty) on turn 4. That same turn, during your main phase, you can immediately activate his [0] ability to create a 2/2 Knight token — no waiting required.

Remember: even though you can act immediately, you still cannot activate a loyalty ability on an opponent's turn unless a card effect specifically grants you priority or another mechanism allows it, since you need priority during a main phase with an empty stack on your own turn under normal circumstances.