The port-side door (next to the dedication plaque) is the turbolift:
Spock exits the lift there when we first see him walk onto the bridge:
When Kirk, McCoy and Uhura race through the ship to warn Pike that there's more than a seismic emergency, they are seen entering the bridge directly from the corridor via the other (starboard-side) door:
When Chekov goes "I can do that"-ing to the transporter room, he exits via the same door.
See also: bridge panorama