Some sources of tap water contains nitrates, so test your water straight out the tap and see if that could be what's going on.
Is your tank stocked at all?
You should test your tap water whether it is suspect or not. It's important to know what your putting into your tank.
Bacteria live on the plants as well. I imagine you took good care of them in transit, therefor all your bacteria survived the trip on them. How about your filter did you possibly take good care of it too.
A cycled tank is a mature bio-filter. Since you carried your bio-filter with you and dropped it into the new tank, than it goes without say that your tank is cycled. But like John K said, be sure it can handle your intended bio load.
As far as rising nitrates, you basically stirred up a ton of ammonia when you stirred your substrate. Your good to go.