No,no,no, that ugly brown gook in the filters is the good stuff!

Whatever you do don't replace your filter media! Now it may be that we'll decide you should gently squeeze out the sponges in tank water (never tap water!) and dunk and swish any loose media in mesh bags or trays. This is you you "clean" a filter. You're just trying to get rid of the loose clogging debris, not the valuable beneficial bacteria that live on the filter media surfaces.
OK, excellent that you happen to already have the right type kit. Now I can trust your numbers enough. Also, its at least nice that PetSmart has Seachem Prime for conditioning, even if they don't seem to carry many other Seachem products anymore unfortunately. Prime is, in my opinion, the best conditioner you can put your hands on.
The most important function of a conditioner (for this function, more or less any conditioner product will be as good as Prime) is to remove any chlorine or chloramines that your water authority has added to kill bacteria in the town pipe system. It should be used in all the fresh tap water that gets added to your tank because chlorine/chloramines are harmful to both fish and plants and of course to bacteria. Prime, though, is also especially good at several extra functions like converting harmful ammonia into harmless ammonium and binding up heavy metals to remove their toxicity. In your situation you should dose prime at 1.5x or 2x the suggested dose but not more than 2x. You should dose per the size water you are adding back after a water change or for the whole tank if you are using a hose to directly fill the tank.
The LFS is incorrect in telling you to use Prime as a way to "bring down the ammonia and nitrate(NO3) levels." The conversion of ammonia to ammonium is only short and temporary and not intended as the way of removing ammonia from a tank. Prime is not used in nitrate(NO3) control at all. The workers are generally innocent but the standard advice in virtually all the LFSs is to "wait a week for any problem and then please come back and buy fish!" I recommend a policy of being friendly and establishing a good relationship with retail establishments but to not take any advice at all, just keep them in good shape for when you need fish and supplies locally.
Did all the water get changed when you transferred the tank to your house?
~~waterdrop~~