Learn Turkish: Çabuk gel.

This is a direct command (the imperative). In Turkish, to tell one person to do something informally, you simply use the bare root of the verb. The dictionary form is gelmek (to come). Drop the -mek, and you are left with the command gel (come!).

Why is there no 'you' (sen) in this sentence?

In Turkish, you can usually drop subject pronouns because the verb gives it away. The bare verb root (gel) is exclusively used to give commands to 'you' (singular, informal). Adding sen to make it Sen çabuk gel is grammatically fine, but dropping it is much more natural.

In English, we add '-ly' to make 'quick' into 'quickly'. Why doesn't çabuk have a special ending here?

Turkish often uses the exact same word for adjectives (describing nouns, like 'quick') and adverbs (describing verbs, like 'quickly'). Because çabuk is placed right before the verb gel, it naturally acts as an adverb meaning 'quickly'.

Can I say Çabuk gel to anyone, like a shopkeeper or a boss?

No, you should only use this bare root form (gel) with friends, family, children, or people you are very close to. If you are talking to a stranger, a boss, or a group of people, you would need to use a more formal ending.