Google Translate is correct in this case.
In English, when we're referring to a future event in relation to something else, we normally use the present tense. Think of it this way: right now, we say the rain will start, because it is in the future. That is, it will start even if we're talking about right now. So it doesn't make sense to say you will be back before it will start, because it already was and has been true that it will start. What you mean is that you will be back before the actual event of it starting - that is, when it starts.