For your first brew, be a little conservative here. Let it go the three weeks. Then check gravity with the hydrometer, then again 2 days later. If same reading and reasonably close to what the kit indicated, then bottle it.
There's a lot to learn in brewing to do it well. It's not rocket science but neither is it simplistic. My advice to new brewers is to stay as simple as reasonable early on in their brewing career. Better to brew a simple beer in an excellent fashion than attempt a complicated beer with process errors.
The more variables you allow, the greater the chance you'll screw one of them up (don't ask my how I know this), and if the beer doesn't turn out, then how do you identify the culprit?
There's a lot to learn, a lot of arcane language and definitions: hydrometer, gravity, tun, mash, krausen, sparge, vorlauf, and on and on and on.
But you know what is the single most difficult thing to learn in brewing?
Patience. Once you get some beer in the pipeline that will be easier, but in the meantime we want to rush the process. While its possible to speed things up, that's not the way to bet, and it is best done with some experience to back up your intuition.
I once did a Kolsch that went from brew day to serving out of a keg in 11 days, and it was a fabulous beer. But that's the exception, and I have some specialized equipment that helped me do that. A 3-week time frame is a more reasonable one for most beers, though there are exceptions on the margins.