Hello; You have the simplest of issues. The media from the established tank should have colonies of beneficial bacteria (bb). Moving some of this media into the new setup will have bb ready to consume ammonia. You should be good to go with respect to the bb and the cycle.
My approach generally is to set up a new tank with the gravel and other stuff I plan to use and fill it with water. I usually let the new tank run for a day with some filtration running and the heaters operational to test the setup for any mechanical problems and get the temperature stable.
once all the equipment and things check out, I usually add one fish and some of the bb loaded material from an established tank. By the way the bb are on the surfaces of many things as well as the filter media. I do this just in case there is some unnoticed issue with the new tank. After from a few hours to a day or so I will move the other fish into the new tank.
I also acclimate the fish going into a new tank pretty much the same way as when bringing fish home from a fish shop. The water in the new setup may be different enough, temp and other parameters, to give a bit of a shock if fish are just dumped in. I have netted fish and dumped them into a new setup usually with no real issues, but from time to time......
Good luck