Learn Finnish: Mä en mee yksityiseen kouluun, vaan tosi kivaan julkiseen. - I am not going to a private school, but rather to a really nice public one.

Vaan means "but rather" or "instead", and it is used to introduce a direct contrast or correction to a negative statement. Since the first part says you are not going to a private school (en mee), vaan provides the alternative. If you used mutta, it would sound like you are adding a completely new, separate thought rather than correcting the first one.

Why are the words kivaan julkiseen in the illative (into) case at the end of the sentence?

They are matching the invisible word kouluun (school). In Finnish, when you drop a repeated noun at the end of a sentence to avoid repeating yourself, its adjectives still have to be in the exact same case. Since the first part was yksityiseen kouluun, the second part becomes tosi kivaan julkiseen [kouluun].

Is mä en mee standard Finnish?

No, it is very common spoken Finnish. In standard written Finnish, this would be minä en mene. At this level, you will see a lot of these shorter spoken forms because they reflect how Finns actually talk in everyday conversation.

How do the dictionary forms yksityinen (private) and julkinen (public) change to become yksityiseen and julkiseen?