HI Froggy and Welcome to Koko's!

Tanks need the fish in to actually cycle. You need a good dechlor like prime, novaqua, amquel etc. Preferably prime really. Spendy, but it doesn't give a false reading with the ammo kits and is concentrated.
There's no reason to use distilled water..... If ph and Kh is low or fluctuates, then simple Baking soda will correct it. The only time ro/distilled waters need to be used is if your tap/well water is really really nasty or you have a salt water tank.
No need for it to sit overnight either, Dechlors take and bind all that.
You should take a test straight from the tap for everything. Then, one from the tank on PH early am and late pm, especially if you have any live plants.......
Drop kits preferable to strips for accuracy!

Ammonia, nitrite,Nitrate, KH & Ph
Now a bit of salt is fine to have on hand, really it can be your best friend and as captk has explained, it can be used medically as well. I do recomend a salt test kit as well. #1 item for the fishy medical kit -salt! That would be rock salt, sea salt, water softner salt 99.5% pure or better even as long as it has no additives/binders such as yps.

Other things you might have use for you will probably already have in your home, like peroxide, iodine, epsome salts....