My personal opinion: Terran suck at islands. Or other way: I suck with Terran on islands. But people ask me to put some Terran island strategy, so here it goes. Be aware that it is not as polished as other strategies posted here...
What to do ...
Number on the left is control. You should produce SCV's constantly.
8 Supply Depot
10 Refinery
12 Barracks
15 Supply Depot
16 Factory
From now on you should produce marines when you have money for them.
When you have 4 marines put a bunker on some strategical position and then continue to build marines
Also remember to build Supply Depots when needed.
Be sure to have money for 2 starports as soon as the factory finishes tho. It should be at 22 control.As soon as starports are finished build add-on on one of them and Wraith from other. When the add-on is ready start cloaking imediatelly. Set new wraiths to patrol. Dont scout with them. When the 3rd Wraith is built you should have money on command center and another Wraith. Build the CC on the edge of your island and when it is finished float it on nearby island. When you have 4 Wraiths go and scout with them. Start Wraith and Dropship in your Starports. With that dropship you should transfer some SCVs to your expansion. Build add-on on your factory. Build Academy + Comsat. Build Engineering bay and some turrets.
When you discover enemy start killing his peons, if you are lucky enough to find a spot without detection. If not, start destroying his buildings and being a general nuisance. If its Zerg kill the overlords. BUT DONT LET THEM KILLED! Their primary purpose is to prevent your opponent from expanding. At home you should buid constant stream of Wraiths + Tanks + Marines + occasional Dropship. Strategy I use with best results is scaring my opponent with Wraiths on the begining and then switching to drops with wraiths only as a support units. It means either 4 Factories and 2 Barracks or 6 Barracks and 2 Factories - depends on if you like more Goliaths or marines.
I dont advise buiding mass Battlecruisers. They consume lots of resources, are very slow to build and slow in movement. I usually use some 3-4 BC's in late game to Yamato defenses and open way for heavy drop. but it is a matter of personal preferences, not a dogma.
Terran has one great advantage on islands - good defense. Islands occupied by Terran are hard to conquer. Bunker + Tank + 4 Turrets can stop huge forces. I tend to add 2 -3 Goliaths in mid game to stop larger drops.
On the picture is early game defense of expansion. Short distance on the left is my main island - that is why the tank is where it is.
vs Terran Notes
You will need some more turrets at your main in case he goes mass Wraiths. I found hard way that vs Terran the best strategy is mass Marine drops. Early game its 2 Dropships + 4-6 Wraith support. 16 marines can destroy lots of things. In mid game I add 3 more Dropships and 3 SVs in the mix for D-matrixing leading dropships and include some tanks. During late game I spice it with few Battlecruisers for Yamatoing defense. I also build comsat immediately on my first expansion. If I know I'm going vs Terran (no random) I build Academy + Comsat earlier.
vs Protoss Notes
Tanks + Goliaths are best combo vs Protoss. I must admit that my primary task when fighting with Protoss is to prevent his Carriers coming in game in mass numbers. I have serious troubles with Carriers as a Terran

. It means no expansion for Protoss as Carrires are close to impossible to support from one base. When I fail that and he is going for Carriers I start building mass Wraiths + Science Vessels. Strategy is: Scan Carrier fleet, EMP it, kill observers, start kiling Carriers. But its difficult...
vs Zerg Notes
Uhmm. Yeah. Well. This is a difficult one when the Zerg goes fast Mutas. Zerg is much faster than you in getting air. When I know I face Zerg I forget this build order and build turret defense first. If I fight random Zerg its race for cloaking. But if you survive go for heavy SVs. Those things with Irradiate just own Zerg.
If he is going double hive and fast drops you are lucky guy. Kill overlords. Hunt them everywhere. And of course go for SVs.
Well, thats about it. If somebody has a better strategy, contact me. I can hold my ground with this, but I am certain that this is not the best thing available

.