From experience, its best to put them in bags instead of straight in the cooler with the water. Its a LOT easer to acclimate them and get them into the tank when you get to your destination when they are in bags.
Test the water at the new location, and basically acclimate the fish just as you would if you were bringing them home from the pet store. As other posters have said, keep your filter media wet.
Set the tank back up as soon as you get to the new place. The fish will be fine in the bags for at least 3 hours (mine had to stay in bags for about 6 and I didnt loose a one.) Theres no need for an air pump, if you use bags, use ziplock ones (I doubled the bags for mine and used the gallon ones) so if need be you can open them to allow fresh air in the bag so the water stay oxygenated.
When you set the tank up at the new place, dont forget to add water conditioner to it, altho you probably wont have to worry about chlorine or chlormaines, there may be heavy metals in the water that your fish wont be used to that water conditioner will help with. When I moved my tanks, I was going to take a good portion of old water with me, but since i had a 75 g , 2/ 29g, a 20g l, and a 37g tank, I was unable to do this, so all my tanks got 100 percent water changes with no adverse affects to the fish.
About the fry. One of my 29g has guppy and there fry in it. There was no way I was able to catch all the fry out of the tank, so.. I caught what I could, left soem water in the tank, removed the majority of the substrate and all the deco. The majority of these fry survived the move, all may have I dunno..i didnt see any dead ones.
HTH