Is a make-up and expiation, or merely a make-up, required for one who missed fasting for two months at the beginning of puberty due to ignorance of the obligation to fast?

What you are obligated to do is make up the days you missed in Ramadan. Ignorance does not waive the obligation to make up these days, as stated in Irshad al-Salik: "Making up the fast is obligatory for breaking it, even if it was due to forgetfulness, ignorance, coercion, or illness." Shaykh al-Uthaymeen mentioned that not knowing about the obligation to fast does not waive the obligation itself, but only the sin. The idea that a Muslim would be ignorant of the entire obligation to fast Ramadan is unlikely, unless they are a recent convert to Islam; in that case, there is neither sin nor an obligation to make up the fast.

If you were ignorant of the impermissibility of delaying the make-up fast, then there is no expiation (fidyah) required for the delay. As for the expiation for breaking the fast in Ramadan, it is to make up the days you missed. The severe expiation (kaffarah mughalladhah) is only obligatory for sexual intercourse during the daytime in Ramadan.