Here is the full list that needs to be complete. A lot of these you'll get just by playing, but the small descriptions should help. If you have any questions, feel free to comment.
*Updated since there were a few down upvotes, went back and added spoilers and some descriptions*
New update 8/4/2020 - Had the wrong islands for the cores, edited)
Touryst Island
- Monument Cores (Story related)
- Picture it! (photo of each monument, one on each island except the one you get from getting all of the bottles and Leysure. This includes the one that
-- Touryst - stand on top of the Towa monument and take a picture of the pressure plates on the little island, thanks Jay Cowl
-- Fijy - Only need the main monument, no pics of the Twins/pedestals used to open
Ybiza
- Let's play (Soccer shirt from shirt store on Hawayy then return and do the soccer minigame)
- Get the fruits (2 melons, 1 strawberry to smoothie maker, they are common around the island)
- Thirst! (Give the smoothie from get the fruits to the elevator person)
- and the beat goes on (finish party meter, then buy records from record store and return them)
Soggy
- Rat Attack (Enter the mine and help clear the rats, required for access to the diamond mine)
- Get licensed (Rain lab at the top of the mountain (not the monument) and take a small quiz)
- The coordinates (Get coordinates after finishing the cores,
Give them to the pilot across from the rain lab)
Santoryn
- Let's go vacation (show the couple the fijy island guide)
Hawayy
- Break the ice! (3 coconuts to ice cream man)
- Surf's up (10k points in surf contest, check 540 degree trick achievement solution for more)
Fijy
- The Sound of Drums (Self explanatory, achievement for beating without missing a beat, replayable)
- The Monument Scrolls (Hidden in breakable walls dictated by cracks then returned to museum, 7 in total)
*** Spoiler - click to reveal *** Provided by Like a Bob Ross
1. Santoryn - On top of mountain behind the club
2. Ybiza - Bottom of monument mountain
3. Fijy - Side of small mountain over from canoe
4. Hawayy - Take canoe to mountain in distance
5. Soggy Island (Mine) - Floor 10
6. Soggy Island - Take canoe to side of helipad island
7. Tyny Island
- Can you do the Canoe? (Talk to uncle inside house to help nephews and do the canoe course)
Leysure Island
- Faces and Phrases (10 pictures of different characters you meet for the art gallery. Must be done in order but you can stock multiple photos at once. Thanks Pickle Rick for the full list)
*** Spoiler - click to reveal ***1. Waiting for a lift - Dave (the guy by the elevator on Ybiza)
2. Show Me Where You Want To Go - Captain (in your boat)
3. Firestarter - Scout (atop Mine in Soggy Island)
4. Live and let dive - Claude (the diver in Santoryn)
5. A man of special taste… - Iceman (ice cream vendor in Hawayy)
6. Smart Reader - Tourist (lad reading a book on the beach in Hawayy)
7. High Scorer - Bob (outside the arcade on Leysure Island)
8. Who is laughing now? - Security Man (at base of pyramid temple on Soggy Island)
9. Hit maker - Drummer (in the band on Ring)
10. These clumsy tourists - Worker (atop the TOWA temple on Touryst Island)
- Pull up! (Purchase tank top from shirt store, return and do 25 pullups in 20 seconds, confirmed that you don't need to do the 35 pullups)
- Break the Highscores! (Purchase tokens in the arcade then beat the top score on the 3 arcade machines)
- Quadcopter Race
Jazz Club (On Santoryn, need tux shirt from shirt store)
- Ella and Duke (Take a pic of the front of art gallery)
Unknown (Island reachable after collecting the 4 messages in the bottles)
- the parcel (Open safe in plane, then return parcel to museum