Be the Best! - Superlatives

If you chose Answer 1, dare we suggest that either you are very rich, or you weren't thinking straight?
Answer 2 is right; Answer 3 would be understood, but isn't quite 'proper English'.
'Cheapest' is not factually wrong, but like many words connected with the big issues in life (money, birth, death, health, sex, home, work etc.) it has 'overtones' ... in other words, it suggests all sorts of other ideas without actually saying them. 'Cheap' is a perfectly good and serviceable word, but we can be sure that most of you would know what we mean by 'a cheap hotel', and even if your money is short, you would prefer not to stay anywhere quite like that.
Sometimes we talk about things being 'cheap and cheerful' (i.e. they don't cost much, they're probably not very well-made or prepared, but they are OK for a short-term purpose, and we can be glad to have them for the time being).
If you watch British television soap-operas ~ to improve your understanding, and maybe pick up some 'cultural' points ~ you may hear one character criticise another for making 'a cheap remark' (= a cruel or tasteless word, easily said, and hurtful to another person), perhaps that they called a woman 'cheap' (which we will not go so far as to explain here; think it through, or look it up elsewhere, if you really feel you need to).
On a lighter note, one of the oldest English puns ( = wordplay, where two words with different meanings sound and/or look the same) is about a pet-shop where they offer 'Baby parrots going cheap' (i.e. you can buy one for only a little money if you want it; but 'cheep' is our English word for the high-pitched sound that tiny birds make when they first open their beaks. Like all good noise-words, it even sounds rather like what it means, where 'the little chicks go "cheep, cheep, cheep"!)
... Not to be confused with Tweets and Tweeting, which (until fairly recently) meant more or less the same thing ...

There is only one other proper superlative here, and that's Answer 2. 'Handiest' is itself a useful word, but it sounds odd to think of a whole network of trains, buses etc. as being 'handy'; we more usually use that word to refer to smaller things ('Keep a pair of scissors handy').

Don't forget the double G in 'biggest' (and 'bigger').
We don't use Answer 1 because 'big' is a short and common adjective; Answer 4 was wrong because it isn't a superlative (there's no tell-tale '-ST' to show that we are looking at the end of a scale or series).

'Nearest from' (Answer 1) obviously won't work; it would be 'nearest to ... ' .
The question mentions 'all these islands', suggesting that there must be at least three of them (otherwise it would have said 'both'), so 'further' won't do, either.
'Farthest' is good, although Answer 4 is also possible. 'Farthest' has a rather solid, historical, almost pompous tone to it ('The farthest place you could travel on earth, in the days of the British Empire, was Australia or New Zealand, and they were both ruled from London anyway.')

For this to make sense, we need the lowest point on the scale (look how the Answers were laid out for you!).
At some point during your dealings with the English-speaking world, you may find yourself completing a Consumer Satisfaction Survey, which might ask you questions such as:
'On a scale of 1 to 5, how satisfied were you with your experience here today? 'Most satisfied / Fairly satisfied / Not sure / Less than satisfied / Not at all [ = least ] satisfied.'
If you know your way down the line from 'most' to 'least', that should be useful for you!

'Best' and 'worst' are the two strangest, but most common opposites in the English language.

This is how we make other such superlatives: 'the best-dressed gentleman', 'the best-designed garden' etc.
Usually these are expressions describing how wonderfully well something has been done to them (e.g. they had really good and thorough preparation, or cleaning, or whatever the process was).
There are reverse, negative forms wherever these make sense ('the worst-dressed / -mannered / -behaved / -spoken', etc.) : 'worst-spoken' refers to someone whose language is poor, in the sense that they pronounce it badly and may well be using a lot of rude words.
Often we attach the two parts of the expression together with a hyphen, though some people prefer to leave that out.
We would be best pleased if you now felt best prepared ~ or, at any rate, better prepared ~ to understand and use such expressions from time to time!

Yes ~ as you saw in the title, we have a short bright single word for 'the most good': 'Best'!

Answer 2 is right because this is the correct way to form the superlative of a long adjective, as we have already seen.

'Most' is fine by itself here; 'greatest', or perhaps 'biggest', would also be possible.
'Grandest' doesn't seem right here (even if you are coming to this from French: 'le plus grand'); 'grand' carries a sense of self-importance and pride. We doubt that the college would want to reward a student who had made such an obvious 'song and dance' (as we say) about how hard they had been working over all that time. Modesty is so much nicer in people!