Is the present continuous necessary in the English translation? "Turkish training starts today" seems more natural. Or at least one of two different interpretations: If the statement is simply informational, the simple present is usually used instead of the continuous present, or even instead of the future: "the course starts today", "the meeting is at five tomorrow" The present continuous is mostly used if you want to stress the currentness of the event, as opposed to other possibilities: "the course is starting now/today/this week" (and not tomorrow or next month)
I imagine this sentence could be part of a conversation between coworkers. Maybe they're having breakfast in the cafeteria. I could imagine that "starts today" might be a bit more likely than "is starting today," but not by much; both are very possible. It's kind of interesting to note that they're talking about a future event while using a present tense. We do this a lot in English when the context makes the meaning clear: "We're going to Mexico!" I tell a friend over coffee in Ohio. "When?" "Two weeks from Friday."
I think it signals that "eğitimi" is part of the noun compound "Türkçe eğitimi."