Great Lighthouse is ridiculously overpowered; in any single-player Archipelago map I'd give it a shot. Colossus is great, albeit not as broken as GLH.
Rapid expansion is generally a good idea. Your limiting factor is most likely going to be how fast you can produce enough settlers, workers, and workboats. Even without GLH, guaranteed overseas trade routes plus coastal tiles if necessary makes it difficult to go bankrupt.
When you've got the hammers to spare, send a workboat/galley/trireme or two out to try and (a) find good islands to settle, (b) meet other civs to trade with, and (c) possibly eventually circumnavigate. Workboats are cheaper, but barb galleys will probably kill them sooner or later so they're a bit more of a gamble.
Slavery is going to be very important early in the game. Odds are you won't have that many hills to work, so you need the whip to use seafood tiles for production. Later on you can rushbuy stuff or use workshops.
Sid's Sushi is extremely powerful late-game; if you can manage to get it and the Kremlin you've won and the AIs just don't know it yet. Even without Kremlin, the population boost can run a ton of specs. to skyrocket your research rate.
Privateers can absolutely cripple an AI if you get them before they have Frigates, and earn enough to pay for themselves in the process (plus they're a good way to earn Great General points).
Late-game, submarine-tactical nuke-transport is by far the easiest way to clinch a military victory - basically all AI cities should be in range of your subs. Defensively, if you have the patience for it, a couple carriers on patrol with fighters can give you plenty of advanced warning of any enemy fleet approaching, letting you crush them before they unload their troops... but it takes a fair amount of micro.
Rather than always using galleys as personalized transports for your units (taking them from where they start to their final destination), consider setting up "ferry" galleys between major neighboring islands in your empire. That way you don't need to sail all the way around those islands every time - you unload on one side, walk across, and load onto a different galley waiting on the far side; it often lets you get away with fewer galleys, saving you a few hammers.
If you have 3+ cities on an island which doesn't have your capital, you start paying colony maintenance costs too. These can get quite expensive, but decreasing distance maintenance also decreases the cap on these costs - so if you have a big island full of cities without your capital, consider putting a forbidden palace or Versailles there (at least be sure to get Courthouses).
Don't bother making an effort to grow to work coastal tiles unless you're FIN. They're only slightly better than break-even for non-FIN leaders (in some uncommon circumstances, they can actually cost you more in upkeep than they earn). If you have the food surplus and nothing better to do with it, go ahead and grow, but don't stress if you can't.
You can use your cultural borders to form bridges between islands before Astronomy. A city's borders will automatically expand to fill it's whole BFC (the set of tiles it's citizens can work) even if some of those tiles are deep ocean tiles. And pre-Astronomy units can enter ocean tiles as long as those tiles are within your cultural borders. So if you have a spot where just 1 tile of ocean separates you from coastal waters on the far side, you can get across that by founding a city near there and popping it's borders once.
Watch out for FIN AIs. They tend to be by far the strongest AIs on Archipelago maps.
Ok, out of Archipelago tips-and-tricks now.