Learn Turkish: Annem bugün alışveriş yapmıyor. - My mother is not going shopping today.

It comes from the verb yapmak (to do). We take the stem yap, add the negative suffix mı, and then the present continuous suffix yor. Because the vowel in yap is an "a", vowel harmony rules tell us to use mı, giving us yapmıyor (is not doing). Notice there is no personal suffix at the very end because it is about "she" (my mother).

The translation says "not going shopping", but doesn't yapmıyor mean "not doing"?

Yes, literally this sentence says "My mother is not doing shopping today." In Turkish, the standard phrase for "to go shopping" or "to shop" is alışveriş yapmak (literally "to do shopping").

Where is the word for "my" in Annem?

In Turkish, you usually show possession by adding a suffix to the end of the noun rather than using a separate word. The word for mother is anne. Adding the first-person possessive suffix -m makes it annem (my mother).

Is bugün a combination of two different words?

Exactly! It combines the words bu (this) and gün (day). Put them together, and bugün literally means "this day", which translates to "today".