The difference between present continuous (presente progressivo) and present simple (presente indicativo) is not as strict in Italian as it is in English. I can say "sto studiando italiano" meaning "I study Italian" or "I'm studying Italian now" with no difference in meaning between the two. But if I were to pick them out, I would say "studio Italiano." It wouldn't have been wrong if I'd used the present continuous.

You can use both "Studio italiano" and "Sto studiando italiano" to mean the whole process. They're both correct. Nonetheless, if you want to split hairs, I'd say that using the present simple tense (presente indicativo) "Studio" is preferable, as it sounds more natural and it's probably the tense a native speaker would use in this case. In fact, generally speaking, the presente indicativo is used to refer to an action that takes place regurarly or a habit (as in English). An Italian native speaker would probably choose the presente progressivo tense (present continous) in a context in which, for some reason, he wants or needs to emphasize that he/she's currently studying or that he/she is still studying. But, in a "neutral" situation, a native speaker would rather say "studio". Spero di esserti stata d'aiuto :)

The difference between present continuous (presente progressivo) and present simple (presente indicativo) is not as strict in Italian as it is in English. I can say "sto studiando italiano" meaning "I study Italian" or "I'm studying Italian now" with no difference in meaning between the two. But if I were to pick them out, I would say "studio Italiano." It wouldn't have been wrong if I'd used the present continuous.