Learn Turkish: Çabuk git ve başla.

To give a direct command to one person (informally), you just use the bare root of the verb. You take the dictionary forms gitmek (to go) and başlamak (to start), and simply drop the -mek and -mak endings.

Why is there no special ending on çabuk to make it mean "quickly" instead of "quick"?

In Turkish, you usually don't need a special ending (like "-ly" in English) to turn an adjective into an adverb. You just place the descriptive word çabuk directly before the verb git to show how the action should be done.

I learned that the verb gitmek sometimes changes to a "d" (like in gidiyor). Why is it a "t" here in git?

The base root of the verb is git. The t only softens to a d when you add a suffix that starts with a vowel (like the present tense -iyor). Since this imperative form is just the bare root with no suffixes at all, it keeps its original t.

Is it natural to use ve (and) to connect two verbs like this?

Yes! While Turkish has many advanced grammatical ways to link verbs together, using the simple word ve between two independent commands is very common and natural, exactly like saying "go and start" in English.