Locations on the picture are labelled with numbers related to locations listed here.
Location 1 (White Chest)
This is right behind Tannis' hut, a short way from the New-U/Fast Travel station.
Location 2 (White Chest)
Head into the structure under the huge radar dish, just to the east of the first chest. The chest is on the north side, inside the structure.
Location 3 (Red Chest)
This chest is down a pathway behind Tannis' hut. The start of the path is to the north of her hut, and the chest is at the very end; after the bend.
Location 4 (White Chest)
This chest is inside the bar, on the other side of the rear counter. To get in, approach the front counter, and jump-crouch. Get out using the same way.
Location 5 (Red Chest)
This chest is behind the high wall that is west of the bar. Go around the wall, and head to the end of the ledge to reach the chest.
Location 6 (White Chest)
Head down the tracks, heading west. The chest is on a circular platform, near where you pick up one of the turret components.
Location 7 (White Chest)
Head all the way west along the tracks, until you reach the end. Jump off to the south, into a small clearing. The chest is under the stairs.
Location 8 & 9 (White Chest & Red Chest)
Head down the path south-west of the spa. You'll reach a structure, and on the other side there will be pipes/bars that you can jump on. Use these to climb up and reach the chests. White chest is near the middle, whereas the Red Chest is at the very top.
Location 10 (Red Chest)
To the west of the building Mr. Blake is in, there is a stairway leading to a balcony with a man on it. Look to the west into the yard below. Jump into the yard to reach the chest.
Location 11 (Red Chest)
This chest is located above the subway-like stairs leading down. There is a Tartarus Station sign with the second "T" pointing to the chest. (Not the sign above the tracks) Jump on the southern railing, then jump again onto the beam. Duck under another beam, and jump across the gap to reach the chest.
Location 12 (Red Chest)
East of Mr. Blake's building is a building with a giant Hyperion sign on it. The chest is at the top of the building. It can be accessed by jumping onto crates and barrels. The crates are on the south side of the building.
Location 13 (White Chest)
There is an alleyway-like path underneath Marcus' store, on the north side of the tracks. The chest is located at the east end of the path.
Location 14 & 15 (2 Red Chests)
This chest is on top of a building two houses east of Marcus' store. Jump onto the curved metal beams on top of the station itself to get to the top of this building. After this chests, jump to the west, staying on the rooftops. Drop down into the fenced area beside Marcus' store and go into the room to get the second chest.
Location 16 (Red Chest)
The chest is located on top of the building just north of Marcus'. There are barrels/crates behind the building, and various things you can jump on to reach the top.
Location 17 (White Chest)
This chest can only be accessed after Phase Four of the storyline. The chest is just through the gate.
Location 18 & 19 (2 Red Chests)
This chest can only be accessed after Phase Four of the storyline. Jump from the platform after the gate to the stone ledge to the south-west. Follow the ledge to the end, then jump north to reach the chest (It's a long jump). Go back to where you started, and follow the ledge to about half way. Jump to the stone landing to the north, and make your way to chest #19.
Location 20 (Red Chest)
This chest can only be accessed after mission It's a Trap... Clap has been completed.
Location 21 (White Chest)
Go to the eastern bridge. Jump down to the stone ledge to the south. From there, jump off to this chest.
Location 22 (White Chest)
This chest is clearly visible from the crosswalk over the tracks. It is near the giant Hyperion sign, in plain view.
Location 23 then 25 (2 White Chests)
This chest can be tricky to get to. It is located behind the center turret, on the southern security wall. Jump down to the cliff from the east side, then walk/jump along the security wall to reach this chest. It is advisable to remain on the wall, and head back east towards chest #25.
Location 24 (White Chest)
The chest can be found at the far west end of the bandit camp, outside of the southern security wall.
Location 26 (18 Red Chests)
Hyperion gift shop's underground weapon storage. Can only be accessed after completing the storyline.
Location 27 (Red Chest)
On top of the southwestern pylon on the far west side of the bridge that leads to Scorched Snake Canyon. Accessible via the cable supporting the bridge once the mission Operation Trap Claptrap Trap, Phase One has been completed.
Location 28 (White Chest)
The chest is on the roof of the building Mr. Blake is in, right next to location 11 on the map.