The direction of the induced εmf is such that it opposes the direction of the change in the original magnetic field. This is where the negative sign comes from in Lenz's Law.
While that's the official answer, I empathize that it might still be confusing. (εmf is endless source of confusion for almost everybody.)
So let me phrase it another way. This way is less elegant, but hopefully more insightful. But first we must make a distinction of our terms.
When you calculated the magnitude, you already made the distinction between the original magnetic field and the rate of change of the original magnetic field. They are two different things, and can be -- and often are -- in completely different directions (and can be -- and often are -- in the same direction). My point here is that making the distinction between strength of the field and strength of the rate of change of the field is an important distinction to make.
Moving on...
You know that a changing field will induce an εmf in the wire loop. That εmf will cause a current to flow through the wire if the circuit is closed (like it is here). That current will induce a magnetic field of its own (through the same loop). And (...here's the important part...) the direction of that induced magnetic field is in the opposite direction of the rate of change of the original magnetic field. I think that's the general rule you're looking for.
