Learn Italian: Prendiamo la metropolitana: è più veloce. - Let's take the subway: it is faster.

In Italian, the noi (we) form of the present indicative is also used as the imperative (command) form to express "Let's [do something]". Context tells us which one it is; since the speaker is making a suggestion, "Let's take" is the correct meaning here.

How do we know più veloce means "faster" and not "the fastest"?

To say "faster" (a comparative), you just use più + the adjective. To say "the fastest" (a superlative), you need a definite article before it, like la più veloce. Since there is no article right before più in this clause, it simply means "faster".

Could I use the adverb velocemente instead of veloce here?

No, because you are describing a noun (the subway), so you need the adjective veloce (fast). You would use the adverb velocemente (quickly) if you were describing an action, for example: il treno viaggia velocemente (the train travels quickly).

Is it common to shorten the word metropolitana in everyday spoken Italian?

Yes! It is very frequently shortened to la metro. Note that even though the shortened version ends in -o, it keeps the feminine article because it is just an abbreviation of la metropolitana.