Right, but do they get to go first in that combat?
No, they dont. Once someone else at the table initiates hostilities by declaring a hostile action, the DM narrates that action commencing and calls for initiative.
On the PC who declared the hostile actions 1st turn in round 1, he can resolve that action, or change it to something else.
Example (Mexican Standoff):
DM: The Orcs stand 30 feet away from you, with weapons drawn, watching you warily. One steps forward to negotiate, speaking to you in Orcish...
Player 1 (Barbarian): Screw this, I charge the Orc and attack him trying to stab him in the face.
Leeeeroy Jenkiniiins!
DM:
Sigh. Ok, the Barbarian screams in rage and leaps forward, with his weapon drawn, charging towards the Orcs, who bellow war cries in return!
Roll initiative.
------Transition from narrative time to combat time has now occurred-----
DM: OK... Player 2 (Wizard) you rolled the highest. You go first. The barbarian has taken a step forward, screaming in rage, and all hell looks like it's about to break loose. What do you do?
Player 2: I cast Mirror image and retreat back to the rear of the party.
DM: OK cool. Next in initiative order its the... Orcs! They race forward to meet the barbarian in mortal combat, meeting his charge with a charge of their own. (Moves each Orc in turn, rolls attacks as needed). Next its... you Leeroy.
Player 1: I enter rage and attack the nearest Orc, so I can hear the lamentations of their women! (rolls)
etc.
That said, a ruling of 'the first guy to declare a hostile action gets advantage on his initiative roll' is cool I guess. It's going to lead to a very trigger happy party though.