Πριν αρχίσω, η αδερφή μου μού είπε να ζεστάνω το τηγάνι καλά.

Πριν can work as a conjunction meaning before (I) …, so it naturally introduces a clause with a verb: Πριν αρχίσω = Before I start.
Πριν από is used with nouns/pronouns (or noun phrases): πριν από το μάθημα = before the lesson.

After πριν (and also after να), Greek commonly uses the subjunctive. Here αρχίσω is aorist subjunctive (perfective aspect), which fits the idea of starting as a single event: before I start (at all).
Πριν αρχίζω would sound unusual in Standard Modern Greek for this meaning.

Here μου is a possessive clitic meaning my, placed after the noun:
η αδερφή μου = my sister.
Greek typically places these possessives post-nominal: ο φίλος μου (my friend), το σπίτι μου (my house).

Normally μου is unaccented. It can take an accent (μού) to avoid ambiguity or to add clarity/emphasis—especially when two clitics might appear nearby or when the writer wants to make the pronoun stand out.
Meaning-wise, μου είπε and μού είπε are the same here: she told me.

It’s a very common Modern Greek pattern: [verb of saying/asking/telling] + να + subjunctive
So είπε να ζεστάνω… literally is “she said (that) I should heat…,” i.e. “she told me to heat…”.

το τηγάνι is neuter singular accusative, used as the direct object of ζεστάνω (I heat).
The article agrees with the noun: το (neuter) + τηγάνι (pan).

καλά is an adverb meaning well / thoroughly. It modifies the verb ζεστάνω: heat the pan well.
Adverbs in Greek are flexible, but placing καλά after the object is very natural: ζεστάνω το τηγάνι καλά.