I have read that it is possible to grow your own yeast, but I have not been able to find exact instructions. I would like to grow my own yeast so that I don't have to spend money on those small sachets, which cost around $3.5 for 7 gr around here.
I haven't actually harvested or grown my own yeast yet, though I really want to. It's about $7.95 per batch for me, and I brew about 3-4 times a month. Adds up for sure.
I've grown my own yeast out of bottles before. Basically, all you're doing is making a very small yeast starter and pitching the dregs of a bottle into it and growing it up from there. I usually start with 25ml - 50ml out of a bottle and then step it up a couple of times until I'm making a 1000ml starter for a batch of brew.
I've had luck doing this with English-style ales and Belgian-style ales, both.
You'll want to make sure that you can sterilize the mouth of the bottle before you pour out of it, since that's the place that's most likely had some sort of other bacteria get to it. You can swab the mouth of the bottle with a small amount of isopropyl alcohol and burn it off for quick sterilization.
A few important things:
1) You must do this from bottle-conditioned beer. If the yeast has been filtered out prior to packaging, there will not be enough yeast in the solution for you to grow.
2) Not all bottle-conditioned beer is bottled with the same yeast that it was originally fermented in. Many breweries use lager yeast for bottle-conditioning, since they can then put the bottles into cold storage or cold shipping and it will still condition, ferment, and carbonate, just more slowly.
3) It has to be fairly fresh beer, as the yeast will suffer quite a bit of stress under packaging conditions (high pressure, not optimal temperature). You may not be able to culture yeast out of something that's been fairly aged, or what you do culture out may be a mutated strain.
4) Over time, yeast strains will mutate. Most commercial breweries will re-pitch yeast batch to batch for a few generations (6 - 10 is normal, I think), but at some point, if you're looking for consistency from your yeast strain, you'll want to wash your yeast and grow a pure culture up from scratch again, and this will be a pain in the ass without some sort of lab. Mind you, in a homebrewing environment this may not be much of an issue, so long as the mutations you get are pleasant.
I'm on my 4th generation of yeasts that I've grown out of bottles, now, and everything still tastes great. I have a yeast library of four yeasts that I've grown up from scratch - two belgian yeasts, one english-style (which I think is Whitbread), and one English-style combination (two bottles of different ales together).
That said, I still buy yeast for probably half of my batches, just for variety in flavor profile.
The first thing you must learn how to do is make a yeast starter. This is simpler than making beer but your sanitation must be very good, don't be intimidated.
Pour a little off the top and taste it before you pitch the starter into your beer. If it tastes off, do not use. Re-evaluate your sanitary technique.
Having a 2-3 liter Erlenmeyer flask is great for this because you can heat it directly and transfer right into a ice bath to cool, eliminating step 4 above. I cracked a flask on an electric burner, so be careful.
See also: Mr Malty's yeast starter FAQ
Now that you know how to grow yeast outside the beer fermenter, there are a few easy things you can do to stretch your yeast. The easiest way is if you buy White Labs pitchable tubes only use half or a third of a tube to make a starter. Spray the cap and threads with sanitizer and re-cap. Try not to get too much sanitizer into the tube.
If you buy the WYeast smack packs you can do something similar with a sterilized mason jar. Boil a quart mason jar & lid for 10 minutes, cool and pour the contents of the smack pack into it.
This is another easy way to get more out of your yeast. Instead of splitting and growing yeast before you make beer, washing cleans trub off yeast you have already used. There is an illustrated post on HomebrewTalk which I will summarize.
When you are ready to reuse the washed yeast, make a starter from one of the pint jars.
Culturing Yeast From Bottles
This is the best discription I've found yet on how to culture your own yeast. http://www.homebrewtalk.com/f163/slanting-yeast-133103/
I also have this book which teaches you handling techniques http://www.northernbrewer.com/brewing/first-steps-in-yeast-culture.html
You can also use yeast from batch to batch, though I'd recommend washing the yeast to prevent over pitching and remove trub. http://www.homebrewtalk.com/f163/yeast-washing-illustrated-41768/