Far Harbor is big. Big enough, and misty enough, that you could miss some of the best bits without even realising they were there. There's a lot to explore by yourself, but this guide will point you in the direction of the coolest things you can find on the island, without spoiling exactly what they are. Go out, and enjoy the delights that the island has to offer! Delights here being a subjective term, of course.
Go to Haddock Cove
Haddock Cove is probably ill deserving of the title 'cove'. It's more of a puddle, to be honest. It's on the east edge of the island, just above where the land starts breaking up into a kind of marshland - if you're helping the Mariner repair the hull in Far Harbor then she'll suggest you go there for mirelurk carapaces, and it'll appear on your map. The location itself is a bunch of shacks and bridges connected by pontoons. As you explore you'll find some holotapes that tell the terrible tale of what became of the Husky Family, who you may recognise if you're familiar with WWE. And we assume at least some of you are.
Get the Vim! Pop Power Armour
Part of the main story in Far Harbor involves collecting some of DiMA's memories and then following down the leads you get from them. One of these will take you to the Vim! Factory, which you should explore carefully. In an office on the top level, on the other side of the mixing vats, you'll find a suit of Vim! Pop promotional power armour. It's pretty stylin'.
Fight a hermit crab
Every so often on the island you'll run into one of these guys. We won't give too much away, but they're probably the coolest enemies in the DLC. Their spawn points are rare, as are they, but you can almost always find one near the Oceanarium, which is on the North-West side of the island. West of that is a broken bridge sticking out into the sea. And on the end of that, if you're lucky, you'll probably find a hermit crab.
Track down Shipbreaker
As you travel around the island you might get a notification that you've picked up Shipbreaker's signal on your radio. It's unclear how exactly it's triggered, but it's probably by helping the residents of Far Harbor with the Turn Back the Fog, and the frequency will be around the area of one of the settlements you've opened up on the island. Tune in to the frequency and follow it until you reach Shipbreaker, who is, it's fair to say, a hell of a fight. Old Longfellow, the grumbly old companion you can pick up in Far Harbor, has been looking for Shipbreaker for ten years, so it's advisable that you take him along with you. When and if you prevail he'll give you a unique weapon, not to mention all the loot you can get from Shipbreaker's corpse.
Investigate a foul and bloody murder at The Cliff's Edge Hotel
You can find it by yourself, but the easiest way to get into the Cliff's Edge hotel is by speaking to Pearl, a Miss Nanny robot who'll be floating around outside the main entrance of Far Harbor, waiting to bring you, the detective, to her superiors to solve a murder. Once she's taken you to the basement level you'll be put to work, gathering clues, evidence, and interviewing potential suspects before finally pinning it on the culprit. This is a slightly dry description which belies how fun and ridiculous it actually is, because we don't want to spoil it. Suffice it to say that you should definitely agree to help Pearl when you speak to her.