If I go to a Language School, do I go to school?

In American English, she is taking classes.

If you were to say 'she is going to school' that would imply that she is a child (elementary or secondary) or attending a college for a bachelor's degree.

Anything else you are either taking classes or going to 'night' school or law school, even if full time and during the day. A post-graduate university degree that is not professional, say for a master's degree in literature is however still 'going to school', but sounds a little funny (like to young).

Actually, 'language school' sounds funny. Even there it would be 'language class'.

So the end results, the most natural thing to say is

How are classes going?

As to what is most natural in British English I'll leave that to someone else.