@user178049 is correct in that "[distinguish] the good from the bad" is an expression. Though you could fit other verbs in there besides just "distinguish." For example, you can take the good and take the bad, and then you've got the theme song for an American sitcom from the '80s.
The point is really that taking out the articles changes the meaning a bit (or at least makes the meaning a bit more ambiguous).
One must learn to distinguish good from bad
is a perfectly acceptable English sentence. But that's not super hard for most people. Little kids generally understand the difference between good and bad, or right and wrong. The trick is applying the theory to the real world.
In the real world, you've got gray areas. You've also got things or people or ideas that might seem good or bad on the surface but are actually the opposite.
That's why we add in the articles. "The good" and "the bad" refer to the specific things in the world that are good and not bad, or vice versa – whatever those things may be. (So they're "specific," but only sort of.)
In answer to your other question about "distinguish," I would say it is okay here. That is just a simple infinitive. You could say "how to distinguish," though; that's okay too.
To me,
One must learn how to distinguish good from bad
implies maybe a slightly different emphasis than
One must learn to distinguish the good from the bad
But I'd say they both get at approximately the same idea.
Both to and for can be used with liable, but they are used in very different circumstances.
Liable to is usually seen as part of the form liable + infinitive, and indicates what someone is likely or probable to do:
He is liable to want a beer with his dinner.
Liable for indicates someone has a debt, legal obligation, or responsibility:
Since he crashed my car, he is liable for the cost of repairs.
In your example sentence, both of them would use "for".
Edit: I've actually now found a number of example sentences in British English with "liable to" used in the sense of a legal obligation, just as with "liable for". So the first one may be acceptable either way in BrE - although as an AmE speaker, I would say that "liable for" is, at least on this side of the pond, the normal phrasing when discussing legal obligations.
Since the original sentence is British English (American English would not use "transportation" in that sense), I will defer to people with more knowledge of BrE on which they would prefer.
Best Answer
I believe that the correct answer would be my having been. The reason for this is that later in the sentence, the author indicates that being a student "has helped me." In order to indicate clearly that being a student occurred before it offered help to the author, you need to use the perfect tense, rather than the imperfect.
see here for reference: http://penandthepad.com/differences-between-past-imperfect-tense-8794443.html