Skyshards are tiny gem-like pieces of Aetherius, that are scattered across all of Tamriel. For every three skyshards you find, you'll earn one skill point. To collect a skyshard, you need to find one and, while standing next to it, press the "interact" button (A in Xbox One, E in PC). A small animation will follow, in which your character will absorbe the skyshard's power. Once the animation is done, you will see a confirmation message. You cannot interact with the skyshard while you're in the middle of a battle, so kill any enemies before. Also note that if an enemy attacks you during the animation, the procedure will be aborted and you'll have to kill the enemies and then try to interact again. But don't tarry, as the enemies respawn!
To unlock this achievement, you don't actually need to find and absorb all the skyshards in all of Tamriel, only those available when the game was released. There are a total of 321 skyshards in the base game, but two of them are not required to unlock the achievement, so you "only" need to find 319. Finding them is actually the challenge. Fortunately, you'll always find one skyshard in each of the zone's delves, and one skyshard in each public dungeon. The remaining skyshards are found in the overland, and varies a lot: at the bottom of a waterfall, atop of a mountain, hidden in a barrel, etc. The in-game zone explorer gives you hints, such as "Secret shrine by Pelin's Church", but honestly your best bet is to refer to a map: there you can look at the skyshard icon, meaning an overland skyshard, and the delve and public dungeon icons (if you zoom in, you can also see the delve or public dungeon map, with the skyshard location.
Below you can read where they are per zone, and I'm including a link to the zone's map (shout-out to UESP's wonderful ESO interactive map, you can start perusing from
https://esomap.uesp.net/ or click in any of the links below).
Daggerfall Covenant:
+ 3 skyshards are available in Stros M'kai island, all of them in the overland.
https://esomap.uesp.net/esomap.html?world=376&x=456055&a...+ 3 skyshards are available in Betnikh island, all of them in the overland.
https://esomap.uesp.net/esomap.html?world=348&x=399902&a...+ 16 skyshards are available in Glenumbra, 9 in the overland, 6 in delves, and 1 in Bad Man's Hallow's public dungeon.
https://esomap.uesp.net/esomap.html?world=363&x=361816&a...+ 16 skyshards are available in Stormhaven, 9 in the overland, 6 in delves, and 1 in Bonesnap Ruin public dungeon.
https://esomap.uesp.net/esomap.html?world=663&x=399902&a...+ 16 skyshards are available in Rivenspire, 9 in the overland, 6 in delves, and 1 in Obsidian Scar public dungeon.
https://esomap.uesp.net/esomap.html?world=579&x=399902&a...+ 16 skyshards are available in Alik'r Desert, 9 in the overland, 6 in delves, and 1 in the Lost City of the Na-Totambu public dungeon.
https://esomap.uesp.net/esomap.html?world=102&x=399856&a...+ 16 skyshards are available in Bangkorai, 9 in the overland, 6 in delves, and 1 in Razak's Wheel public dungeon.
Ebonheart Pact
+ 3 skyshards are available in Bleakrock Isle, all of them in the overland.
https://esomap.uesp.net/esomap.html?world=619&x=399902&a...+ 3 skyshards are available in Bal Foyen, all of them in the overland.
https://esomap.uesp.net/esomap.html?world=166&x=399902&a...+ 16 skyshards are available in Stonefalls, 9 in the overland, 6 in delves, and 1 in Crow's Wood public dungeon.
https://esomap.uesp.net/esomap.html?world=644&x=399902&a...+ 16 skyshards are available in Deshaan, 9 in the overland, 6 in delves, and 1 in the Forgotten Cripts public dungeon.
https://esomap.uesp.net/esomap.html?world=287&x=399902&a...+16 skyshards are available in Shadowfen, 9 in the overland, 6 in delves, and 1 in Sanguine's Demense public dungeon.
https://esomap.uesp.net/esomap.html?world=602&x=399902&a...+ 16 skyshards are available in Eastmarch, 9 in the overland, 6 in delves, and 1 in Hall of the Dead public dungeon.
https://esomap.uesp.net/esomap.html?world=318&x=429688&a...+ 16 skyshards are available in The Rift, 9 in the overland, 6 in delves, and 1 in The Lion's Den public dungeon.
https://esomap.uesp.net/esomap.html?world=698&x=399902&a...Aldmeri Dominion
+ 6 skyshards are available in Khenarthi's Roost, all of them in the overland.
https://esomap.uesp.net/esomap.html?world=137&x=399902&a...+ 16 skyshards are available in Auridon, 9 in the overland, 6 in delves, and 1 in Toothmaul Gully public dungeon.
https://esomap.uesp.net/esomap.html?world=124&x=399902&a...+ 16 skyshards are available in Grahtwood, 9 in the overland, 6 in delves, and 1 in Root Sunder Ruins public dungeon.
https://esomap.uesp.net/esomap.html?world=401&x=570801&a...+ 16 skyshards are available in Greenshade, 9 in the overland, 6 in delves, and 1 in Rulanyil's Fall public dungeon.
https://esomap.uesp.net/esomap.html?world=435&x=399902&a...+ 16 skyshards are available in Malabal Tor, 9 in the overland, 6 in delves, and 1 in Crimson Cove public dungeon.
https://esomap.uesp.net/esomap.html?world=510&x=399902&a...+ 16 skyshards are available in Reaper's March, 9 in the overland, 6 in delves, and 1 in The Vile Manse public dungeon.
https://esomap.uesp.net/esomap.html?world=547&x=399902&a...Other
+ 16 skyshards are available in Coldharbour, 9 in the overland, 6 in delves and 1 in Village of the Lost public dungeon.
https://esomap.uesp.net/esomap.html?world=203&x=399902&a...+ 15 skyshards are available in Cyrodiil, in the territory controlled by the Daggerfall Covenant, 9 in the overland (2 of them within the Covenant headquarters) and 6 in delves.
+ 15 skyshards are available in Cyrodiil, in the territory controlled by the Ebonheart Pact, 9 in the overland (2 of them within the Pact headquarters) and 6 in delves.
+ 15 skyshards are available in Cyrodiil, in the territory controlled by the Aldmeri Dominion, 9 in the overland (2 of them within the Dominion headquarters) and 6 in delves.
https://esomap.uesp.net/esomap.html?world=252&x=232666&a...Please note that the other two skyshards: the one obtained at the end of the quest "Soulshriven in Coldharbour", and the skyshard found in atop of White Fall Mountain in Cyrodiil (within Pact territory) do not count towards the achievement (although they will earn the player in-game achievements).
All the skyshards located in the three alliances are relatively straightforward, and come by naturally by doing the alliance quests, then Cadwell's silver and Cadwell's gold, as long as you explore and do the delves and public dungeons. The maps should be enough to locate them. The skyshards located in Cyrodiil are more difficult to earn, because the PvP mechanics of the area will allow other players to kill you, in which case you'll have to respawn at an alliance-controlled keep or outpost. This, coupled with the fact that Cyrodiil's map is much bigger than the alliance zone's, can make it a bit cumbersome and tedious. Following is a list with general recommendations regarding Cyrodiil skyshards.
+ Select a campaign that has fewer players. As of May 2020, there is a campaign for under level 50 players, as well as a campaign with CP points disabled, both tend to have the least amount of players in them. The less encounters you have with enemy PvP players the faster you can get those skyshards.
+ Cyrodiil map is huge, and it can take you a long time to move from one keep to the other. While the transitus shrines will help, they will only take you so far, so you'll have to ride a lot. Therefore, it is recommended that you max out your ridding skills, both speed and stamina, at least to 50 and 10 percent, respectively.
+ Consider acquiring the skill "Rapid Maneuver" from the Alliance War / Assault skill line, as it will increase your horse speed for a significant amount of time.
+ If you're a PvE player, chances are you don't have the gear to face PvP players. PvE gear tends to favour magicka/stamina and critical hits, whereas PvP gear favour health/magicka/stamina balance and spell/physical penetration. This means you're likely to die in a duel. Therefore, do not engage enemy players! Do not walk into enemy holds (or resources like farms and mines), instead try to move around even if it takes longer to reach your destination.
+ Carry some invisibility potions (you can craft it with Nirnroot + Blue Entoloma at any alchemy station). Some classes, like the Nightblade, have invisibility skills. If an enemy player spots you, drink the potion / cast the skill. This might give you a chance to escape.
+ Travel through the country side and avoid the roads, especially those connecting enemy keeps, since there's a higher chance to encounter enemy players.
+ Cyrodiil's delves are more complex than the ones found in the alliance zones. Most of them have two or three bosses, instead of the one, and the mobs are more difficult (akin to public dungeons). Be prepared with health potions, just in case. Note that if you're killed by the mobs an no enemy player is around, you can revive normally with a soul gem stone. If you do defeat the delve bosses at the same time as you hunt the skyshards, you'll earn the Cyrodiil Cave Delver achievement as well.
+ Be mindful of the map! Many delves and skyshards are deep into enemy territory, which will make it harder for you to get there unspotted and untroubled. When you see your alliance controls a portion of territory usually in the hands of the enemy, do take advantage (this is specially true for the three alliance-controlled villages: Bruma, Cropsford and Vlastarus, since there will be guards in town that might attack you if they're controlled by enemy alliances). Be aware that even if your alliance controls the entire map, it won't last much more than a few hours. Eventually the enemy will regroup and take back their territory.
+ Pay attention at which keeps or outposts are the enemy alliances attacking or defending at any given time. This means that players from the nearest keeps will be moving to and fro, so you'll have a bigger chance to be spotted by groups of enemy players.
+ As an enemy alliance controls deep territory, the players are likely to spawn in the furthest keeps or outposts, to be closer to the action. This means that the skyshards and delves found near their headquarters will be relatively unprotected and unpopulated. Take advantage of this!
Finally, a special advise on the four most difficult skyshards to obtain. Each alliance has two headquarters: Northern and Southern High Rock for the Covenant, Northern and Southern Morrowind for the Pact, and Western and Eastern Elsweyr for the Dominion, all of them located on the corners of the map. These headquarters are protected by special gates: Ni-Mohk protects Northern High Rock, Alma Ruma protects Southern High Rock, Gate of Chim protects Northern Morrowind, Ghartok gate protects Southern Morrowind, Altadoon gate protects Western Elsweyr, and Mnem gate protects Eastern Elsweyr. Each headquarter has a skyshard nearby. Two of them will be in your alliance headquarters, so those are the easiest skyshards to obtain in Cyrodiil. But the other four will be near the enemies' headquarters, which means you'll have to pass over those gates. There's a catch though. Not only are they well guarded by NPC guards (70k health each, and will overwhelm you easily unless you dispatch them quickly), but the gates can be locked, meaning you cannot pass even if you defeat the guards. These gates cannot be unlocked using siege weapons. The only way the gate opens is when the alliance that owns the gate loses control of the nearest castle/fort/keep. For instance, if the Dominion or Pact captures Fort Warden from the Covenant, then the Ni-Mohk gate will open. Conversely, if the Covenant captures Fort Warden, then the Ni-Mohk gate will close. And so on for all the gates (A message will pop-up in screen whenever a gate is opened or closed). This creates a special situation in which you need your alliance (or the opposed enemy alliance) to actually control any of these castles! There's another obstacle: neither of these keeps are needed to crown an emperor, so PvP groups are not as likely to attack them. You can always join a PvP group and convince them to attack those keeps, but you'll probably have to attack the other keeps as well. You can wait out until a group of your alliance or of the opposing alliance captures the keep (perhaps motivated by them wanting to capture an Elder Scroll located in the temples past the gates), and then make your move, but this approach can take a lot of time. Unfortunately those are the only options you have. Once the gates are opened, passing by is not that difficult, although there's a risk you might get killed. So, first, make sure no enemy players are lurking by. Then you can try stealth with invisibility potions to try to sneak undetected by the guards. Or, if you have ward spells and high health, or high stamina, you can simply just run by: the guards won't follow you for long. Avoid the roads and head to the skyshard, possibly using invisibility potions/spells, if you have any, and circling around if needed to avoid detection. After you have got the skyshard, repeat for the other three gates and you're done!
Each time you collect all the skyshards for a zone, an in-game achievement will pop-up (like "Glenumbra Skyshard Hunter" or "Coldharbour Skyshard Hunter"). Once you have all the alliance zone's Skyshard Hunter achievements, you'll get another achievement (like "Dominion Skyshard Hunter"). As for Cyrodiil, each time you collect the skyshards belonging to one alliance, you'll earn another achievement (like "Cyrodiil Pact Skyshard Hunter"), and once you earn the three, you'll earn the "Cyrodiil Skyshard Hunter". Once you have all the zones', all of Cyrodiil's, and Coldharbour's, you'll earn the Tamriel Skyshard Hunter achievement.