Learn Russian: Раньше я жил здесь, а теперь там.

You are right! Раньше literally means "earlier" or "before." In Russian, saying "Earlier I lived here" is the most natural way to express the English idea of "I used to do something."

Why does the sentence use а instead of и for "and"?

The word а is a linking word used to show contrast, often translated as "whereas" or "but." Since we are contrasting the past (раньше) with the present (теперь) and "here" (здесь) with "there" (там), а is the perfect fit. The word и is only used when adding similar things together without contrast.

What is the difference between теперь and сейчас since both mean "now"?

Теперь is used specifically to show a change in a situation, meaning "now as opposed to before." Since this sentence contrasts the present with раньше (earlier), теперь is the most natural choice. Сейчас usually just means "right now" or "at this very moment."

Is a word missing in the second half of the sentence? It just says "and now there."

Just like in English, Russian can drop a verb if it is already obvious from the context. The full idea is а теперь я живу там (and now I live there), but you do not need to repeat the verb. Notice that if you did repeat it, you would have to change it from the past tense (жил) to the present tense (живу).