Learn Finnish: Kurkkuni on kipeä.

Yes, it does! In Finnish, the word kurkku means both throat and cucumber. You have to figure out which one is meant from the context. In this sentence, having a sore throat makes much more sense than having a sore cucumber!

Where is the word for "my"? Why isn't it Minun kurkkuni?

In standard written Finnish, the possessive suffix -ni at the end of kurkkuni is enough to mean "my". You can add the pronoun minun at the beginning for emphasis (Minun kurkkuni on kipeä), but it is not required.

Does the word kurkku change at all when adding the -ni suffix?

No, it doesn't change here. You simply take the basic dictionary form kurkku and attach -ni directly to the end to make kurkkuni (my throat).

How would I change this sentence to say "Your throat is sore"?

You would use the second-person singular suffix -si (your) instead of -ni. The sentence becomes Kurkkusi on kipeä.