Not sure about the medication part, someone else will have to help with that.
As far as not screwing up the cycle, that's easy (presuming the tank's not too large). (Warning, because you are dealing with a lot of organic material, this is going to smell. I suggest getting an air-freshener for your car for after the move)
This all needs to be done just before the move, to minimize the stress to your fish.
Take the filter media out and put it in a ziplock back filled with tank water. Set this in an ice-cream bucket (Just in case the bag leaks)
Take the fish out and put them in whatever you'll be using to carry them (will be determined by the size of the fish. Plastic bags for smaller fish, perhaps a cooler for a larger fish like an oscar) along with quite a bit of the aquarium water, as well as air at the top of the bag.
Drain most of the water out, leaving just enough to keep the gravel wet.
If you have live plants, they, of course, need to be kept at least damp if not fully submerged.
I would suggest keeping all decorations damp, too, maybe by pushing them into the substrate a bit, as they are going to also have the beneficial bacteria on them.
Transport everything, set the tank up as quickly as possible, making sure to get the water temp pretty good to start out with (err on the low side, as heaters, in my experience, tend to do their work pretty quickly) Don't forget water conditioner.
acclimate your fish, then reintroduce them.