Learn German: Ruf mich bitte vorher an. - Please call me beforehand.

This is the informal singular command (imperative) form of the separable verb anrufen (to call). To give a command to someone you address as du, you take the normal present tense form (rufst) and remove the -st, leaving just Ruf. As with all separable verbs in a main clause, the prefix an is pushed to the very end of the sentence.

Why do we use the accusative mich here instead of the dative mir?

The verb anrufen always takes a direct object, which requires the accusative case. Even though some other common communication verbs in German take the dative (like antworten or helfen), anrufen is strictly accusative. So you will always hear mich anrufen, dich anrufen, etc.

Since our current lesson is about the conjunction nachdem (after), how does vorher fit into our time vocabulary?

vorher is an adverb meaning "beforehand" or "previously." Unlike nachdem (after) or bevor (before), which are subordinating conjunctions that connect two separate clauses and require specific tense sequencing, vorher just sits inside a single clause. Its direct opposite is the adverb nachher (afterwards).

Can I change the word order and put bitte at the beginning of the sentence?