Requiem
The first terminal in the game can be found on the second mission (there isn't one on the first mission). At the very end of the level, you'll find a couple of Hunters fighting against some Sentinels. Kill the Hunters and then you'll enter a Forerunner structure. Walk down the long hallway. Towards the end will be a ramp leading up to a large elevator and under the ramp will be your first terminal.
Forerunner
After Rally Point Bravo, you'll fight through quite a few Prometheans. After some fighting, you'll come to an area where you'll be walking down some gravel with a ramp that leads up to a Forerunner platform. Under this platform (the ground will lead down towards it) will be another Terminal. It is pretty visible but it will blend in pretty well with its surroundings.
Infinity
After a jog through the woods, you'll come across some allies. Chat with them a little bit and then you'll be asked to clear an LZ. Immediately after this cutscene, you'll find yourself in a long hall with a ramp leading down in front of you. Turn around and you'll find the terminal right behind you.
Reclaimer
You'll find this terminal after meeting the Librarian. As soon as you regain control, you'll be attacked by a pack of Crawlers. Kill them and then grab Cortana from the nearby pedestal. Take stock before leaving because below you will be a couple of Knights. If you're low on ammo, grab a Suppressor from one of the dead Crawlers and then make your way down the ramp. You can use the large rails attached to bridge for cover until you get to the central structure. Clear out the Knights and Crawlers here and continue through the door ahead. When you go through the door, you'll come to an intersection. If you turn right, you'll see a Terminal. Make sure to use it before taking the Elevator down and leaving.
Shutdown
Take Rally Point Delta and fight your way through the first set of Hunters by the elevator. Kill them. Before taking the lift, there's ammo on either side of it. Stock up and head back to the previous tower. Turn right and go to that tower. If you go around the far side of the tower and look towards the tower, you should find a Terminal. Catch the lift when you are ready.
Composer
After meeting up with the Doctor on the station, you'll take an elevator down and be let off in the same room with the Composer. Follow the waypoint as if you were to leave this room but don't. Walk up the ramp towards this waypoint and when you reach the door to leave the room, turn right and walk off the platform. You should find a small cave just large enough for you to fit inside. At the end will be the terminal.
Midnight
After Lasky makes a hole for you, you'll be on foot with a rather large bomb strapped to you. Walk forward a little ways and you'll enter a room with quite a few Watchers in it. There is a "dead" Turret in this room towards the back and if the Watchers can get to the point where it was, they can rebuild it and cause additional trouble for you. It is likely that with so many Watchers, you won't be able to prevent this so be sure to hang back if you fail to stop them. In the back middle of this room should be more ammo. When you are ready, head into the next room on your left.
This room is going to have a Knight Lancer. If you have your SAW, short controlled bursts will make very short work of him. The door behind him (on the right side) is your next destination and there will be another Knight on the far side. Handle that Knight the same way and go into the room. One of theKnights likely dropped a Scattershot. If you are low on ammo, feel free to grab it. Take out the remaining Watchers in this room and any Crawlers that appear after that. Before leaving this room, go around the back of the central structure hanging from the ceiling to find the final Terminal.
