Corvettes are probably the easiest answer. You can research them quickly and cheaply, but they still require Level 2 naval bases and level 1 arms industries. If you're a Security Council member, you can set your ships to aggressive fire, and they will fire on any enemy ship (including troop transports) that comes into range. Keep them in shallow water near your port cities. The best way to deal with stacks of infantry coming to your island nation is to never allow them to step foot on land. Sink them in the water before they disembark. If you have fire control, you can set your artillery to fire at approaching enemies as well. If you get to submarine production, position your subs strategically in deep water shipping lanes that lead to your country and set them to aggressive fire control as well.

If you don't have fire control and you're asleep,... I guess the best thing you can do is to just build up what infantry forces you can in your cities and maybe build bunkers? Ships set to aggressive are going to be much more effective, though.

I would also go for corvettes.
I´m not 100% sure, buit I think if you put them direclty on the shore, they will lock invading forces in combat.
(the moment they finish disembraking is the moment when war is declared, in this moment the corvette is still in contact and locks )

@Fereyd that is exactly what happens, I use it almost every game with great effectiveness

nope

You don't need a corvette for every city. Post corvettes to defend some, and double up troops in the others. Remember that your opponent starts out with limited troops too. They can't send large stacks to seven cities simultaneously. You can put corvettes in some and outnumber them in others.

As far as I know (sow discussion here in forum about same issue), war will be declared when enemy forces already unloaded on your lands, so any navy stationed near you port cant lock enemy units, they are able open fire to unloaded enemies only , but this is navy against land units already. So "corvettes blocking" tactics dont work.

Here is no effective islands defense for offline time sadly. Even if you build bunker, place there lot of infantry and destroy invading enemies, you will loss your city's buildings, population and morale.
Best what I can do if I want defend owned islands in first game days is build fast some navy (of course corvettes are faster way, but frigates might be ready 1 or half day later only and arent resources wasting as corvettes) and try to patrol most danger sea routes, you may also use initial fighter you have for this purpose. If you find any neutral forces, moving to your side, just destroy them at once by navy.

does this corvette locking work on multiple units trying to disembark?

say, the enemy transport ship sees a naval contact, assumes its a ship, which it is, and then splits his units into two stacks, the first one being just enough hp to keep the boat attacking it while the second slips thru?

or does it lock on all transports passing by

Ground units in every port city/province; corvettes parked right next to those same port cities/provinces. Throw in some bunkers and/or military outposts with some long-range artillery, and your island essentially becomes so high-cost to take that people tend to leave it alone until later in the game. But you can't just sit around and wait for them to come back at you. You need to have naval units ready to counterattack your attacker's coast cities, with maybe a sub or two off each province port. Also, don't discount some Lev. 3 rockets. When stacked, those guys are a nice little addition to any coastal defense.

Also right answer: you cant defend islands from early rusher selfmurder headhunter especially...