Archipelago/island maps, while harder from an overall conquest strategy, are set up for surgical strikes. While I am still a noble level player, here are some tips I have thought up. I will probably organize them better and add them to the article that is in the sub-forum
Speed rules. Circumnavigate the world, get refrigeration tech and if possible give your slow vehicles (tranports, aircraft carriers and battleships) flanking I and navigation promotions. at least the transports. with the greater movement range, the enemy will not be able to flee. The enemy may land troops near you, but only once, since they have no units left to retreat to home ports.
Armored vehicles and flight are not that necessary. get these after all of your naval and infantry/marines techs.
Never land your troops if you have to. this keeps them safe from "thinning" by enemy artillery.
Artillery cannot bombard a city from a transport. this is ok since you have the navy to do this for you. give the artillery barrage and city raider promotions, do not give artillery the accuracy promotion. Cannon will die a horrible death against infantry in a city when attacking from a transport. Use artillery, and they still will die a horrible death, but not quite as horrible.
Navy seals with combat 1 and pinch can take down fortified infantry with city defense I and II provided that you have removed all of the culture bonus and softened the up with anywhere from two to four artillery. The first strikes will keep the seal alive and with decent health. Since marines/navy seals have march, they can be used to island hop really well. give one marine medic 1 per transport, and you are fine.
my troop config is: 2-3 transports marines/seals, 1 tranport artillery, 1 tranport infantry. Try not to leave marines or seal behind in the city, that is what the infantry are for.
There are no borders on an islands map. Yes there are culteral boundries, but these are inconsiquential even for the first attack. to knock a civ out of the game, you probably do not have to take more than 3 cities, the capital and the biggest commerce and production city. If you are going to be on the offense, spy or caravel the target sites, preposition your troops so you can at least start removing the culteral defense of the city the first turn you declare war. Once you take the top three cities within 2-4 turns of declaring war, the AI will hole up since the other cities will barely have the troops to defend themselves and no production capability to get better.
Remember that this can work against you also. Defend well your important cities since you will not have the capability to get the ammount of reinforcements as you would in a land game.
Since your island is a natural born commerce city and food cities, production is difficult. I have gained new respect for workshops playing on the island map. In one of the games I played, my capital was my only strong production city, so I gave it both the heroic epic and ironworks national wonders. pumping out 3 units every four turns is sweet and the AI will not be able to keep up. Slavery is your "gaming" friend. Pop-rush settlers, banks universities or what ever, your city will recover just fine. When unversal suffrage is available, then you can "buy" any wonder after about 3-5 turns at zero science without much of a tech hit. Universal suffrage also gives you production boost from your cottages!
Happiness and health are a big issue since your cities get huge. In fact, war can cripple your empire due to the cry baby citizens singing that hippie song. jails, mount rushmore, theatres, coloseums and the culture slider at 20%. I would not recommend police state since universal suffrage/slavery is the only way to build on one-square island cities and slavery is out once you can get caste system or emancipation.
Wonders are nice but not necessary. If you get lucky with stone or marble or are industrious, then go for them, else take them from your enemies with selective city attacks (see above). I think it is better to build the troops, navy and infrastructure and take what you want, since you then have both the wonder and a sizaeable force.
I will try to organize this better and add it to the naval article in the sub forum. I need to stop goofing and get back to work