I generally go from grain to glass in about 17-20 days, depending on the beer. As was already mentioned, pitching the proper amount of yeast at the proper temperature means a healthy fermentation.
Generally, fermentation will last about 5 days for most medium strength ales. Give it another day or two to make sure, and then dryhop (if doing so) at about day 12. Then keg in 3-5 days.
Using a yeast strain that clears the beer well is a big help- a strain like S05 never clears for me, or at least in my time line, and I don't like it anyway so I use a similar but more flocculant WLP001 in many American style beers. I also like Wyeast 1272 for IPAs and APAs, and it clears fast.
Brewing techniques like getting a great hotbreak and cold break and taking care of fermentation means that the beer is ready to go pretty quickly.
Beers that have complex flavors or really high ABV (like a Belgian triple) take longer so I have those in the mix and allow them to age while I'm drinking the others.
To carb up the keg, I place the keg in the kegerator and put it on the gas at 30 psi for 36 hours then purge and keep it at 12 psi for the rest of the time. It's pretty nice in two days, and highly drinkable in four days. So I'm usually drinking my APAs/IPAs/milds/ambers/Irish red/etc at between day 17-20.
I don't cold crash (except in the keg), and I never use gelatin or other finings. They aren't needed in my case I guess.
Most well made beers don't need time to age except for a few exceptions- barleywine, tripel, lagers, big stouts, etc.
If you are trying to age out off-flavors, the secret there is to not create them in the first place.