Having brewed for about 8 years now, I've finally developed a bit of patience.
I take my hydro reading on brew day, as it goes into the fermenter, just to see how close I am to my target. Then it sits in the corner, in my fermentation chamber for 3-4 weeks, longer for my heavier beers. Then, I usually rack to a secondary for about 2 more weeks, again, heavier beers - longer. I never take another hydrometer sample during this whole time.
When I'm finally ready to bottle or keg, I take the reading - I've never had to re-pitch yeast, (perhaps I'm lucky, but perhaps I'm just giving the yeast time to do it's job). My final gravity is usually close to the predicted value, and it tells me my ABV. Then into the keg or bottles for 3 more weeks before I even think about sampling. 2 more weeks before I share with anyone.
yes, that's about 8 weeks minimum (for me), from brew day to drinking...
so, how soon should you? who knows.... but waiting and temperature control have served me well.
Welcome to your new hobby!