Learn Icelandic: Fyrst dregur hann spilið.

This is because of the V2 (verb-second) rule. In Icelandic, the main verb almost always wants to be in the second position of a sentence. Since the sequence connector Fyrst takes up the first position, the verb dregur must come next, pushing the subject hann to the third spot.

I learned the verb "to draw" as draga. Why does it change to dregur here?

Draga is a strong verb, and many strong verbs have a vowel change in the singular present tense. Here, the root vowel a shifts to an e to form hann dregur (he draws).

Is spilið in the accusative case in this sentence?

Yes, it is! The verb draga forces the object being drawn into the accusative case. Since spil (card) is a neuter word, its accusative form looks exactly the same as its nominative (dictionary) form. The -ið on the end is the attached definite article meaning "the".

Does the word fyrst only mean "first" in a sequence?

In this specific context, it means "first" as step one of a process. However, you might also see fyrst used as a conjunction meaning "since" or "because" in other sentences. Here, it is setting up a sequence of events, which will likely be followed by words like svo (then) or loksins (finally).