Have you arrived on the island? NO? Then get out of here, go back on the road to Sunset Isle! 
ON SUNSET ISLE
I won't go into too much detail here, just explore the island for yourself and have fun! Go shopping for instance; there are all kinds of shops here, and some of them have items that aren't available elsewhere. There's also a bookstore, and some people have books that are unique to the island - you could be reading for hours! You can visit the bars, they cater to different tastes: from the seedy Black Pot to the classy Smoking Pipe, with a restaurant upstairs and entertainment downstairs, to the artsy Tester's Lair. Umrahel's Place is Members Only and unfriendly to strangers, but I'm sure you'll find a way to join. If you're more criminally inclined you could break into some houses that are locked - I'm not talking about the "doors that don't budge", your lockpicking skills will get you nowhere on them - and steal (or just look at) expensive artwork. Or just walk around like a tourist. ("How interesting!")
TRANSPORT ARRANGEMENTS
Your stay on Sunset Isle can be very pleasant, but you might want to (or have to) go elsewhere sometime. There are several ways to travel to Vvardenfell, by ship and through teleportation.
There are several teleportation options, but none of them provide a quick return option, except Mark & Recall. There are no holdings of the Empire or the Temple on the isle, so Divine Intervention will take you to Wolverine Hall in Sadrith Mora; and Almsivi Intervention will take you to Molag Mar, which is the closest Temple site.
If you have visited the Arcanum Tower, you may have found Baboo in the Arcana Halls. She can send you to Vivec, the Foreign Quarter Plaza; Tel Aruhn; the Grazelands (between Falensarano and Indoranyon); and Molag Mar.Another alternative is a mage called Noomo, he's in a room upstairs in the Black Pot. He can send you to Ebonheart, which can come in handy if you are
OPENING THE TRANSPORT ROUTE TO EBONHEART
You may decide to take a ship back to Vvardenfell. You could use the ship that brought you here, but being stranded somewhere outside Tel Branora is no one's idea of fun, I think. However, there is an alternative. On a ship near the Black Pot you'll see someone on deck called Conradus Rex, he might take you. But he's hesitant... When you turn around you'll see someone familiar standing onshore - Betelgeuze! Whether you like him or not, it's more advantageous to take up his offer and visit Myth in Ebonheart, so you can use his ship next time.
There are several ways to get there. You could leave it for later and wait until the main quest takes you to Vvardenfell, and combine your travels
efficiently. Or you could make use of the services of Noomo described above. Talk to Myth in Ebonheart, he's still in the same place where you saw him
first. He wants you to switch some documents in the chest in Canctunian Ponius' office, in the East Empire Company Hall (near the statue of the
dragon). This should be pretty straightforward if you know how to open locks and remain unseen.
Give the right documents to Myth, and he'll declare the transport route open. Go to the harbor and take a look: on the left you'll see Conradus Rex' ship, anchored next to a smaller ship. Go on, talk to Conradus inside the captain's cabin, he'll take you back to the island. And back to Ebonheart again, if you want.
ANGEL QUITA'S BELT
There's a noble lady in the rich part of Sunset Isle who's in need. Go to the west end of the Isle, and you'll see a building upstairs called High Hall. Go inside, and speak to Angel Quita; she's at the other end of the room. At first she probably won't mention any quest, but if you compliment her enough to get her disposition over 80, end the conversation and then start a new one, she'll tell you her little secret. Apparently she had a belt that was stolen from her, and she needs a tough hero (you) to go get it for her. She gives you a key that gives you access to Doovers, a house in the port area, next to the Black Pot.
Once you enter that house you'll notice the strange atmosphere there. There seems to be nobody home downstairs, there are only some chests with valuables (that you can happily take). But upstairs, when you open the door, you'll see some Hunger-like creatures, two big ones and two small ones, and they'll attack you on sight! The Argonian mage who's also walking around there doesn't even acknowledge your presence though. But that doesn't matter, you'll be too busy fighting the creatures for the time being. After killing them you can talk to E'vaarda, the mage, and she'll be quite pleasant. But once you take the belt you see lying on the floor (Angel Quita's belt!) her attitude changes around completely!
But of course you can defeat her too. Return to Angel Quita and give her the belt, and she'll give you a handsome cash reward of 5500 gold.
THE EXPERIMENT
People warned you about going to the Arcanum mages, and maybe NMR tried to talk you out of going to the Arcanum Tower, because they think there's some shady business going on. But sooner or later you'll go there anyway, so I might as well tell you about it. 
In the top of the tower, Heaven's Domain is the domain of a mage called San Cornelius Svantesa IV. He would like to use you for, erm, ask you to join in an experiment. You have plenty of opportunities to back out and run away, but of course you're the proverbial cat with more curiosity than is good for you.
The experiment is about making you a so-called "Shade", which is like an undead version of you. It has its advantages, but they come at a price...
To start the process of conversion you'll need to get five portions of vampire dust and two of bonemeal. Unless vampire hunting has been a part of your career and you kept some of the trophies (the dust), you have your work cut out for you now. You could get the bonemeal easily enough, No Mere Ranger sells it in her shop. But she won't have any vampire dust - it's about the only ingredient she doesn't have for sale...
But most adventurers are avid collectors, I heard, so chances are that you'll have at least some of that dust tucked away back home somewhere. Therefore it shouldn't take too much of your time, aside from travelling.Take the ingredients back to San Cornelius, and without further delay he'll turn you into a Shade, or at least the first stage of it, a half-Shade. Some of your skills, abilities and resistances will have improved, but that is balanced by your new vulnerabilities - daylight and fire! It's not too bad though, but you're not a full-strength Shade yet.
To complete the process (or reverse it if you have second thoughts) you need to get the heart of a young lich, a certain Merius. You can find him in a Daedric ruin underneath the city. If you did the ritual of cleansing, you may remember the door with a lock of 50 in the underwater maze. That's one way to get into the ruin. But another way is through the prison complex, and that's where I'll take you now.
Go into the prison and walk all the way to the back, ignoring the side passages. You'll find a heavily locked door, but of course there is a key for it, and the nearest guard has it! You can try to pickpocket him or kill him, then take the key and open that door. Follow the passages through two rooms, the last one (with a pool of water behind it) has a hidden door to the left. Go through it and the door behind it, and you'll enter a cave.
It's time to go swimming now. Jump in and go norht, you'll find a dark hole you can crawl through. On the other side you can surface again, in the
Underground area of Sunset Isle. When you encounter another pool of water, there are several routes you can take - go right (the left passage takes you to the underwater maze) and then left again.
Now is the time to sneak or fight. You'll pass a cave on your right side, with two bonelords, one of which is a Demon Bonelord. They're just there, you don't need them. A little ahead is a Dremora Lord, but watch out: it will use fire, and you're much more vulnerable to that now. So be quick.Then you'll get wet again and arrive at a junction. You could go left to fight some Daedroths, but your goal is straight ahead. Upstairs there's a nasty opponent, a Demon Skeleton that really lights the place up. He 's your last opponent before you meet Merius, the young lich.
Merius is not as tough as the guy you just killed, but using something with fire damage would be a good idea (as you know from your own experience now, heheh). Then take the Black Heart from his corpse and go back to the Arcanum Tower.
San Cornelius will give you a choice: you can become a full-strength Shade, or be returned to your normal state as it was before the experiment. A very generous offer, I think. Make your choice wisely, it's the last chance you get. Once a full Shade, you can never go back to your old self again, like a vampire could. Besides, the only one who knows about this process of conversion dies at your feet if you go through with it.
Now look at yourself. (I'm assuming you're a full Shade now.) Your new abilities can be very advantageous to you, if you know how to use them. They can even make your progress through the main quest a lot easier, enemies will be afraid to fight you, and you can go anywhere unnoticed. Sometimes people won't even see you when they stare you in the face! You just need to avoid daylight, then you'll be fine.You can use all means of transport that are available on Sunset Isle, even the ships (but not on Vvardenfell!). You can also kill with one touch, if you have enough magicka.
Have you had enough of looking around on the island, at least for now? Then go to the Cathedral.