When Doing the security missions on the first island, you will have to build an ACU center. When it's built, press A on it and select the helicopter and take control of it. Fly the helicopter close to your dinosaurs and press Y to pull out the gun. Just get close and shoot any dinosaur in the face to unlock the achievement.
When the dinosaur goes to sleep, you get this achievement:
Hit a dinosaur with a tranquilizer from 125+ meters for this achievement:
Tranq a dinosaur that is running for this achievement: