最好还是问问老师吧？还是最好问问老师吧？ Which is correct? and Why is it? 这个问题，最好还是问问老师吧.

这个问题，还是最好问问老师吧.

It means "Sure enough, the best is that we ask teacher about this problem."

Which is correct? and Why?

Both of them go well?
There are few previous posts that addressed the adverb order. You can have a search and take a look. 

For natives, we determine the order by intuition, which is formed by using/speaking the language for many years. 

Back to your examples. Your first sentence, 这个问题，最好还是问问老师吧, is more natural and idiomatic. The second one, 这个问题，还是最好问问老师吧，is not that natural, but if you put it into practice, especially given that you aren't a native, it's still acceptable. 

Anyhow, like other languages, we have more tolerance in colloquial speech than writing.