Я учился русскому сегодня is the way to go or Я учил русский сегодня. The latter is colloquial and most likely what an average speaker would say. The former is more of a grammar nazi thing, so kind of bookishly formal I suppose.

No, you don't have to add язык to clarify your point, but you still can do it. Totally up to the speaker. Well, if you need to be all formal, then язык is required