Congrats on the new fishie!
Just running the tank for a week will not make it safe for fish...that's an old myth. You of course want to run it for a day or two to make sure everything works but after that you have to cycle the aquarium. Do a Google search for "fishless cycling"...you'll get plenty of websites.
See, fish produce ammonia through their gills and waste. Ammonia is highly toxic even in small amounts. Luckily there are beneficial bacteria that "eat" the ammonia and convert it to nitrites. Nitrites are also toxic but eventually another (and more sensitive) group of bacteria will show up and convert the nitrites to nitrates. Nitrates aren't harmful to fish in small amounts but can cause issues if they get too high. Live plants will use up some nitrates as food but you'll have to mostly keep them down by doing water changes (and vacuuming the gravel in the process). I hope that makes sense
. So basically, cycling is letting colonies of beneficial bacteria stabalize before adding fish. They need a food source (ammonia), though, so you need to use fish food or something else that will produce ammonia. By adding some fish food to the fishless tank every day it'll rot and create ammonia. Then the bacteria will start to appear and cycling will begin.
You can use aquarium test kits to moniter ammonia, nitrite, and nitrate levels. Your tank is fully cycled when there are no traces of ammonia or nitrites. This can take several weeks in a new tank.
There are ways to speed up cycling
. You can take filter media or gravel from an established tank (one that is cycled and has been running for a while) and add it to your tank. The good bacteria live in the media and substrate so this way you are adding a colony to your tank. Sometimes tanks will cycle in days this way. Live plants also speed up the cycling process.
There is one way to instantly cycle your aquarium. You can use a product called Bio-Spira. It's tricky to find and needs to be kept in the fridge. Most likely you'd have to order it online but it's well worth it. Bio-Spira is live beneficial bacteria and you add directly to your filter (or the bio-wheel of your filter if you have one). You add fish right after adding it and the tank cycles overnight without any major spikes. I've personally used it to cycle three aquariums. Oh, and there are lots of similar products on the market but they don't do much. So don't waste your money on them.
The usual place I order Bio-Spira from seems to have disapeared and so has my back-up place, lol. But apparently Drs Fosters & Smith have it. And they are reliable. Setting up a new aquarium: Ammonia Removal: Marineland BIO-Spira
I hope this all helps.