"I decided to go to the island."
Translation:Jeg besluttede at tage ud til øen.
There appears to be no way to know whether to use 'sig for' after besluttede. In one sentence - "the man decided.." sig for is used. In this one - "I decided - sig for is omitted. In the English sentence, the verb - decided - is exactly the same and so is the usage, so I can't see any way that you can get this right without guessing!
Yes, "tage" + preposition is a very common way of saying "go to". Think of it as saying "I take myself to", but without saying "myself". This one would be something like "I decided to take myself out to the island". The used preposition depends on the speaker's position compared to the destination. Since an island is way "out" there, when you're standing on the shore, "langt ude" then "at tage ud" is used. Another example is"Vi tager ind til byen", often heard in the suburbs "We take (ourselves) in to (into) the city". Others are "over til", "op til", "ned til", "hen til".
This is difficult, so please ask for clarifications if you're still confused! :)