Learn Polish: Gramatyki uczy mnie ten nowy korepetytor. - This new tutor teaches me grammar.

This is the required structure for the verb uczyć (to teach). It takes two objects: the person you are teaching must be in the Accusative (mnie), and the subject matter being taught must be in the Genitive (gramatyki).

English says 'teaches me grammar', so why isn't mnie in the Dative case here?

Because Polish verbs don't always align with English grammar. While you might think of it as teaching grammar 'to' someone (which would imply the Dative case), uczyć strictly requires the Accusative for the student. You are directly 'teaching the person', not 'teaching to the person'.

I thought mnie was the Genitive form of ja. Is it doing double duty here?

Yes! For the pronoun ja (I), the Genitive and Accusative forms are exactly the same: mnie. Even though it looks like the Genitive form you already know, it is functioning purely as the Accusative object in this sentence.

The sentence starts with the object gramatyki and ends with the subject ten nowy korepetytor. Why this unusual word order?

Polish word order is highly flexible and used for emphasis. Putting gramatyki first establishes it as the topic ('As for grammar...'), while the subject ten nowy korepetytor is placed at the very end because it is the newest, most important piece of information being introduced to the listener.