1) Know when your beer is done. Take a final gravity reading(s). If you become familiar with a yeast and your process you will begin to know where your final gravity will end up. Leave the beer on the yeast for an additional 24-48 hours for it to absorb diacetyl and other undesirable compounds.
2) Use yeast strains that ferment quickly. I use English ale yeasts (S-04, WLP002) almost entirely and consistently get 3 day fermentation. Ensure you are oxygenating well to reduce lag time. These yeasts also help carbonate quickly as well. I have had beers fully carbonated in less than a week(they still benefit from longer conditioning time, still)
3)If you really want to get serious about it, start kegging. This will cut down your carbonation time considerably. You can get drinkable beer in an hour or realistically a day or two.
*This is all for ales, disregard if you are producing lagers.