Thanks for all your help everyone. I just have a few more questions:
When I am ready to move my fish, can you explain the process step by step so I don't run the risk of harming, stressing, or killing my fish. I have 1 bag that I kept when I bought my fish. I was gonna get another so I can move my fish all at once since I was gonna take some of the water out of the old tank (which is 30 gallons). If the paramters are the same between both tanks (Ph, ammonia, nitrite, nitrate, temperature) I'd just net them into the new tank. You could also put them in a bucket with some of their current tank water and then add a 1/2 cup of the water from the new tank into their bucket every 20 minutes or so until the water volume in the bucket doubles. Moving them twice might be more stressful instead of just putting them into the new tank though unless something is really off in numbers between the two tanks.
Also, should I any cleaning after the tank is cycled and I'm doing the water change. The tank walls and decor are covered in white powder from the coral being in there. You can wipe the powder off with a glass cleaner sponge (sold at most fish stores) and then suck it up with a siphon during water changes. Other than that, just keep up with a good water change schedule (50% weekly is recommended) and regular substrate vacuums.
How long should I wait after changing the water should I move the fish? You mean after changing the water in the new tank? You can do it right away if levels in both tanks are the same as I noted above.
And my final question is how soon after changing the water and adding fish should I change the filter cartridges. Or should I replace it before I put the fish in? No, don't replace them now! You'll lose your bacteria. I wouldn't touch the filters for another month or two unless something happens to them. After that you can swish the media in old tank water once a month during water changes to remove debris, etc. Don't replace the filter cartridges unless they are falling apart and even then you'd want to replace one portion at a time if possible.