Welcome! I stumbled upon this niche indie game and had to give it a go, It's amazing!! When I saw this didn't have a guide, I knew I needed to do it justice. I hope this guide can help others have the same lovely experience I did :)

For ease of use,
When I say "residential islands" I mean the following:
Green Island
Busy Island
White Island
Volcano Island
Cave Island

Beat story mode

Completing the story mode can be done by getting to the square island leftmost on your map that reads "???", then walking to the middle. Make sure to stock up on food and water by going to the other islands on the way! It's a long journey.

Catch 50 fish

A fishing rod is available on all residential islands to buy for 5 gold.
Gold is a coin can be found in the ocean, dismount your boat and swim there in order to obtain it.

Fish suffice as food, just like fruits you find on islands. They scurry around in the sea, when using the pole, place your bait in the path of a fish. When they bump into it, click to reel them in.
Repeat 50 or so times.

Become friends with a special someone by talking to them a lot

There is a bartender, who looks similar to the achievement icon, on most residential islands inside the bar (which looks like a long house from the outside). To "talk" (or I guess listen) to them, continuously buy drinks from them until there is no option to do so anymore. As you buy drinks they will open up to you about themselves and their struggles with a new dialogue option each time you have one.

Repeat this for 2-3 islands you travel to, and this achievement should pop as you get their final dialogue.

Note: It costs one gold per drink, so make sure you've saved at least 10 or so per island to reach the limit. If you don't have enough, you can always swim to find more and come back later.

Spend 10 soul points on magic

Soul points are the magic particles scattered across water. When you see one, like gold, swim towards it to collect it. When you unlock a magic skill, you can press Q to choose which one you'd like to use one of your soul points on to trigger the effect. See how to find magic skills in Master Magician.

Do this 10 times to get the achievement.

Find a magical island only visible to those with deep souls

Soul Island can be found by collecting around 50 soul points. See Novice Magician if you're confused on how to get soul points. When you get there, the achievement should trigger as you're greeted with the unlock of another magical skill.

If it spawns on top of your boat, beaching it, you can swim away to make the island go away.

Learn all the magic skills

Magic skills are used by going to the small islands on your map labeled, conveniently, "small island". You need 5 soul points to unlock one, do so by talking to the person in the middle.

There are seven skills to unlock, therefore, you must go to seven small islands to get all the skills, costing a total of 35 soul points. Make sure you grab soul points on the way so you have enough when you get there.

Beat random mode with more than one players

Go to "Random Mode" in the main menu. Put the player count to anything from 2 - 4, and advance to the ??? island. This is significantly harder than single-player random mode, due to the fact you need to ration food between two or more people, when the total food isn't increased due to more players and also because if one player dies, everyone must start over from a save.

You can achieve this by either using Steam Remote Play to invite your friends, or, controlling two players at once using different control schemes.

Beat random mode with only one player

Go to "Random Mode" in the main menu. Put the player count to 1, and advance to the ??? island. This is easier than the multiplayer counterpart due to the fact that if you lose hunger, thirst, or energy you only faint and wash up on a nearby island instead of having to start over.