Whenever a creature enters the battlefield under your control, it explores. (Reveal the top card of your library. Put that card into your hand if it's a land. Otherwise, put a +1/+1 counter on the creature, then put the card back or put it into your graveyard.)
Path of Discovery’s triggered ability triggers along with any other abilities that say that the creature explores when it enters the battlefield, including abilities that come from the creature itself or from multiples of Path of Discovery. You may take actions between each resolving ability’s exploration.
Once an ability that causes a creature to explore begins to resolve, no player may take any other actions until it’s done. Notably, opponents can’t try to remove the exploring creature after you reveal a nonland card but before it receives a counter.
If no card is revealed, most likely because that player’s library is empty, the exploring creature receives a +1/+1 counter.
If a resolving spell or ability instructs a specific creature to explore but that creature has left the battlefield, the creature still explores. If you reveal a nonland card this way, you won’t put a +1/+1 counter on anything, but you may put the revealed card into your graveyard. Effects that trigger “whenever a creature you control explores” trigger if appropriate.