After pouring off a beer from a bottle conditioned variety I want to harvest, I'll leave the dregs in the bottle and recap or plug it and put it back in the fridge. It's best if you can collect up a few of those bottles. When you're ready to propagate, make a small, low gravity starter (about half cup 1.030 wort), let the bottles warm to room temp, and then pitch them into the starter. Put it on the stir plate, then slowly ramp up the volume of starter over the course of about a week. It usually takes a while to get an adequate volume of yeast, so the first time I pitch that into a recipe, I'll usually start with a smaller volume of wort (2-3 gallons.) I'll then harvest enough healthy yeast from the primary to use in larger batches.
It's work and it can be time consuming, but it's rewarding. I'm now using a Bell's strain as my house yeast, which all started a year ago as the dregs from one little bottle of their beer.
Note that the yeast from some breweries is already available commercially, so you can save yourself some time depending on what beers you are trying to clone.