Using liquid yeasts, making starters and cultivating yeast from a bottle.

When you get into brewing a bit you start to realise the importance that yeast plays in the process of producing good beer. It seems obvious but when you start with kits you're presented with a simple packet of dried yeast to pitch on top of your mixed up wort and away it goes, no calculations, no messing with starters, just pitch it and go.
This simple dried yeast approach will work and will produce you good beer but it can be taken so much farther and apart from anything else no one has really managed yet to produce a good lager yeast in dried form.
Yeast can impart a huge amount of character and flavour to a beer, especially unfiltered homebrew, so much so that I am beginning to dislike filtered commercial beers as they are simply missing too much character.
I started with liquid yeasts by simply throwing a ladle or two of yeast from the last brew in onto the next brew, there is great advantage in reusing your yeasts for a few brews, it kinda hits the ground running and gets to work on your beer straight away and there is less lag time to the start of fermentation. Less lag time closes the window to infection as your wort is at it's most vulnerable before the yeast kicks off and you want to make this time as short as possible to exclude infection by wild yeasts. You also have the advantage if you are brewing a house beer of allowing the yeast to get to grips and "grow-up" with your recipe. Commercial breweries reuse their yeast strains for years by yeast banking, that said they employ a small microbiology lab with -80c freezers to cryogenically store their yeast strains. Every few generations they test the yeast for strain drift and wild yeast infection, regrowing from their frozen yeast bank and starting again whenever the need arises, but with proper sterile technique they can continue with the same iteration over multiple brews. This is not advised past about 8 generations as a homebrewer for the aforementioned sterile technique that most homebrewers are not very good at.
So at its simplest just throw some old yeast on top of the new brew from the bottom of the last brew. You can also get into yeast washing, I don't tend to bother with that, but if you are interested in this technique I'll point you at a very good post on HomeBrewTalk, a thread by Bernie Brewer on yeast washing.
Some people will talk about acid washing.....it's a last ditch attempt to save a yeast strain and not something that needs to be done on a homebrew scale, so if you want you can research it yourself farther.

It is rarely a good idea to pitch a new beer on top of an old yeast cake as it's almost always over-pitching but again I'll defer to a rather brilliant and extensive post on the subject by a professional brewer on HomeBrewTalk who says it better than I ever could.

I've recently taken the liquid yeast thing a few steps farther and have begun using the White Labs liquid yeast strains. So far I've used the Irish Ale yeast WLP004 and also the Budwar lager WLP802 strain to great effect.
I used the Irish Ale yeast on my Tiger Blood stout
and the Budwar strain I used in my Bi-Winning Lager. The stout turned out absolutely delicious with the yeast certainly bringing a lot of "Guinness" character to the brew. The jury is still out on the lager as it is in its seventh week lagering right now and not yet bottled.
I have recently acquired some lab equipment and some agar and plan to start my own yeast bank based on yeast slants shortly, I'll blog about it when I get that far.

On to making starters, there are a few things necessary for this, most of which you have already and some which would be nice to have, the nice to haves are stir plates and Erlenmeyer flasks, but for the simplest starter all you need is a beer bottle or a two litre coke bottle, depending on what size starter you're making and also what yeast source you're starting with.
The yeasts I mentioned above are delivered to you in quantities that are meant for direct pitching to wort without making a starter, that said because I brew such large volumes, 60ish litres, this is almost three times the size of the average 23l homebrew beer volume so I tend to grow up starters.

This is what the White Labs pitchable stuff looks like, that said, this is not as sent to me originally, it is harvested lager yeast from the brew I did, I just reused the container to keep it, it has been washed with water as described in the post I linked to on HBT.

It's said that homebrewers tend to under-pitch yeast by a factor of up to ten compared to a commercial brewery, so if you want to up the quality of your beers then look after and pitch enough yeast.
Using a Wyeast liquid propagator pack such as the one in the following picture means that you must use a starter to get the desired results.

Wyeast also do a pitchable quantity which they sell as Activator packs, with the Wyeast range you have a small pack inside the larger pack which you need to smack, they are called smack packs for this reason, and when you pop the inner membrane it releases a starter liquid into the yeast sample and the pack swells up, when it's fully swollen is when you are supposed to use it.

Liquid yeasts have limited viability, although they are pretty resilient, but on the pack is a production date and you can use the mr malty yeast calculator to find out how many packs you need and the size of starter you need to grow for a given volume, mr malty is very handy for any yeast calculations and I'd tell you go and bookmark it if you're going to brew as you will find it very handy.

As always with yeast sanitation is very important, I've started to use star san solution mixed up using the battery acid top up water from Halfords as it's deionised and I reuse the mixture again and again and check the pH now and again, I may do a post on sanitation in future and I'll go into this in more detail.

Equipment
Decide on the size of starter you're making and pick an appropriate vessel for the size.
Yeast sample you want to propogate.
Malt Extract - 10g per 100ml of water ( if you have some spare wort from a brew this is even better)
Yeast nutrient. ( add as per instructions)
Thermometer
Kitchen pot
Hob
Weighing scales

Clean your equipment well and sanitise it as you normally would. Put your chosen volume of water in a pot on the hob and then add the appropriate amount of Malt Extract, this is 10g of Malt to 100ml of water to give you an OG of around 1.030 which is good for starters. I don't boil starters as I don't think there's any need, but I will heat it up to 70c for at least 5 mins to sanitise it.
If you only have a small sample of yeast such as when you are propagating the yeast from the bottom of a bottle, Coopers yeast is great for this, then you need to start with a small starter of 100-200ml, add some yeast nutrient, then grow it for a day or two and then step it up. You can start even lower than 100ml, if you have a very small amount of yeast as in the bottom of a bottle where you don't have a thick layer on the bottom but only a small amount of yeasty beer then you can go as low as starting with a 20-30ml starter, just step it up as previously.
Basically add your yeast to the starter wort and then cap it with a piece of sanitised tin foil. Put this somewhere around the same temperature as you are going to brew your beer, shake the vessel every few hours when you think of it, ideally a stir-plate would keep it sufficiently aerated and wouldn't need to be shaken by hand.
If you're stepping up then pitch the grown starter in a day or two into a litre or two litre vessel of wort and wait again.
Some people would pitch a starter whole, well if you're using the same wort to grow it as you are pitching into then you can do this, I use Dried Malt Extract or DME so I'd rather pour off the starter wort and just pitch yeast sludge. You can stand the starter in the fridge overnight and it'll drop all the yeast to the bottom and clear the starter and then just pitch the sludge.