Root currently features 37 achievements, 16 of which require the game's variuous DLCs to unlock. Most of them are rather straightforward, meaning you should unlock them simply by playing for a while and grinding a little bit.
In this guide I've included a couple of basic tips for the slightly more complex achievements, along with clarifications for those with somewhat vague descriptions. Any DLC required for specific achievements is also listed.

Unless otherwise stated, all achievements requiring you to perform specific actions a certain number of times are cumulative across all games you play. All achievements can be earned in regular matches either online or when playing versus AI, but not in local multiplayer games. Actions performed during challenges also seem to count towards at least some of the achievements.

Instead of unlocking right away, achievements gained during a match will be unlocked afterwards, as the screen announcing the match's winner comes up. As far as I can tell, an achievement won't be unlocked if the action that would have triggered it also caused a player to win and thus ended the game.

Craftsmen
Craft 50 times.

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Wrecking Crew
Destroy 2 enemy buildings in a single battle.

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Sneak Attack
Eliminate all attacking warriors with an Ambush.


Once drawn, you can hold onto an Ambush card, allowing you to use it as when another player attacks you in a matching clearing. Given they can't counter with an Ambush card of their own, you're then allowed to remove up to two of the attacking warriors.
Thus, you'll have to use an Ambush card to successfully counter an attack of either one or two warriors in order to obtain this achievement.

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Dominance Victory
Win a game using Dominance.

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Coalition Victory
Share a victory in a Coalition.


Coalitions are created when a Vagabound plays a dominance card and see said Vagabound team up with the player who's got the least victory points. To go for this achievement, either play as the Vagabound or fall behind in points during a game with a Vagabound, before making a comeback.
As the Vagabound, you will no longer be able to score points and instead need to support your coalition partner. Improve your relationship to give them cards, attack the other players to prevent them from scoring, wipe clearings to enable your partner to move/build there etc.

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Challenger
Complete all challenges.


This includes all of the base game challenges along with the harder version(s) unlocked by beating them. All of the required challenges are found under the "Standard"-tab, with none of the other ones added through updates and expansions counting towards the achievement.

Grand Decree
Have a dozen cards in your decree at once.


Try to always add two cards when expanding your decree, using a bird card whenever possible for more flexibility. Through this, build a balanced decree with easily manageable actions; lots of move actions are especially handy.

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Iron Grip
Complete an entire game without entering Turmoil.


When adding to your decree, always chose to only use one card to keep things manageable. Stay flexible by adding as many bird cards as possible and going for easy to perform actions, while keeping your decree balanced. The number of roosts you can place in particular is very limited, so add at most one build action. You'll mostly have to play things save, even if that means losing the game in the end.

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Eyrie Emperor
Win 25 games as the Eyrie Dynasties.

The Mightiest Army
Have 25 Warriors on the map.


For this achievement, you'll simply need to have all of your warriors on the map at once. When playing with the Riverfolk Company, make sure they don't have any of your warriors in their funds.

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Workaholic
Overwork two turns in a row.

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Duke de Cat
Win 25 games as the Marquise de Cat.

Collector
Have at least one of each kind of item at once.


You'll need to collect at least one bag, boot, coin stack, crossbow, hammer, sword, torch and root tea. Each Vagabound starts with a torch and is provided a boot, a bag, a hammer and a sword from the four ruins. The coin stack, crossbow and root tea need to be crafted, gained from another player by aiding them or found in the Vagabound's starting equipment.
In regards to the last option, the Thief begins the game with a root Tea, the Ranger and Scoundrel start with a crossbow, the Vagrant holds a coin stack and the Harrier has both a coin stack and a crossbow.
The Vagrant and Scoundrel are part of the Riverfolk Expansion, while the Harrier is included in the Exiles and Partisans and Vagabonds DLC.

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Socialite
Improve your relationship with 3 different factions in one game.


Simply hand at least one card to each of the players; reaching the first relationship level is enough. Remember that you won't be able to improve your relationship with players after you've attacked them.

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Vagabond Commander
Command allied warriors to move or battle.


The Vagabound gains the ability to move or fight alongside another player's warriors placed on the same clearing the Vagabound is at by improving their relationship with said player three times. Attacking the player at any point will prevent you from improving your relationship further.

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Bodyguard
Assist 10 players in battle as the Arbiter.


This achievement heavily relies on other players assisting you by making use of the Arbiter's ability. You'll essentially have to try to position yourself in clearings with potential for battles and hope. Aside from that, try chatting with other players in multiplayer rounds to encourage them.
The arbiter is included in the Riverfolk expansion.

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Dirty Rotten Scoundrel
Remove 10+ pieces in a single action with the Scoundrel's Scorched Earth ability.


Simply use the ability on a clearing on which there are ten or more tokens, buildings or warriors in any combination.
The Scoundrel is a part of the Riverfolk expansion.

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Vagabond King
Win 25 games as the Vagabond.

All Your Base
Have all three bases on the map at once.

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Revolutionary
Destroy 5 or more enemy pieces in a single revolt.


Trigger a revolt on a clearing which holds five or more enemy tokens, buildings or warriors in any combination.

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Liberator of the Woodlands
Win 25 games as the Woodland Alliance.

Faithful Crusader
Crusade 15 times.

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It's All Connected
Perform 3+ Conspiracies in a single round.

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Cult Leader
Win 25 games as the Lizard Cult.

Corner the Market
Establish a Trade Post in every river clearing.


This includes only the clearings with access to a body of water. The autumn map has five such river clearings, the lake and winter maps each have four and the mountain map has just three, making it the easiest for this achievement.
The lake and mountain map are part of the Underworld Expansion.

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Prosperous Peddler
Sell 100 services.

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Company Man
Win 25 games as the Riverfolk Company.

Kaboom!
Remove 5 or more enemy pieces with a single Bomb Plot.


Flip a bomb plot on a clearing which holds five or more enemy tokens, buildings or warriors in any combination.

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Plotter
End a turn with one of each Plot type on the map.


You'll need to place at least one bomb, extortion, snare and raid plot on the map. The plots don't have to be placed in a single turn and it doesn't matter whether they're flipped or not.

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Grand Conspiracy
Win 25 games as the Corvid Conspiracy.

Lord of all Lords
Have all three lord ministers in play at once.


You'll lose one of your highest ministers whenever one of your buildings is destroyed, meaning to get this achievement you'll have to prevent any building destruction for at least two turns. You should group your buildings on only a couple of clearings and keep those well protected. Besides this, do your best to ensure you can actually sway the lords by influencing your hand cards and/or ensuring you have tiles on clearings matching your hand cards' suits.
Avoid going for this achievement in games with factions that can destroy your buildings directly without counterplay, such as the Lizard Cult or the Scoundrel Vagabound.

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Underground Express
Move 5 or more warriors from the Burrow to the map in a single move.

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Underground Supremacy
Win 25 games as the Underground Duchy.

Amphibious Warfare
Start a battle in a clearing you just moved to with the Ferry.

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Ferrier
Use the Ferry 20 times.

Mountaineer
Clear 20 closed paths.

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Towering Success
Score the Tower for 3 turns in a row.

I hope this guide could assist you in getting all achievements. Make sure to tell me in case I messed something up or didn't make things clear enough at any point.

Also, consider rating the guide and/or dropping an award if it helped you. It’d be much appreciated.