Firstly you need a container to keep your composting worms in. You can buy ready-made wormeries, and many of these are very good. Or you can find instructions on the Internet to make your own. The things you need to remember when choosing or building a wormery are that worms are living creatures and have certain requirements. Your worm bin will need drainage and air holes, but you also need to ensure that the worms can't escape and that they're kept in the dark.
Source your worms. You need proper composting worms - also called red wigglers or brandlings. You can buy them mail order, or get them from fishing tackle stores as they are also used as bait.
Composting worms are used to living in a composting environment, and eat their way through their own bodyweight in waste every day. Regular earthworms don't like being kept in composters - they like to build burrows underground and remain undisturbed.
When you've got a suitable wormery, and your worms, you can set up the bin. Add the worms into the bin, with some suitable bedding. If you bought your worms mail order then they may have come with a coir block that you can soak in water and break up. Otherwise you can use shredded paper or something similar, but make sure that it's damp. Use something organic, so the worms feel at home. It will compost down with the rest of the waste, but that doesn't matter.
Once you've settled the worms, close the lid on the wormery and leave them to get used to their new home for a few days. Resist the temptation to keep lifting the lid - they don't like the light and you will disturb them. Once they feel at home they are less likely to try and escape from the bin.
You can start to add your kitchen waste a little bit at a time. The worm population will gradually increase to match the available food supply, but at the beginning it's important not to add more food than they can handle. Too much food will start to rot and smell bad - if that happens, stop adding new waste and let them catch up.
Worms love vegetable foods - peelings, tea bags, coffee grounds, even kitchen paper and coffee filters. They don't like meat, dairy products and fish. They're also not fond of spicy foods and too many acidic foods, so don't add too many onion skins or citrus peels.
Once your wormery is up and running and you're adding your kitchen waste, you need to make sure to drain the resulting liquid off on a regular basis so that the worms don't drown! The liquid run-off is very nutritious, and when diluted (to the color of weak tea) with water, it makes a great liquid feed for houseplants or garden plants. You'll get much more liquid feed from your wormery than compost.
It will be months before you need to worry about removing finished compost from your wormery. The worms will tend to move out of the finished compost and into the newer waste, and you can separate them out when you remove the compost, and then re-start the bin. You don't have to remove all of the worms from the finished compost - they're native species and will help improve the soil if they get loose in the garden. The population in the wormery will soon recover.
Vermicomposting is a very easy and efficient process. Once you've set up the bin and settled the worms, very little maintenance is required. You should check on the worms at least every couple of weeks to make sure they're happy in their environment. You may find that the pH sinks too low (the bin becomes too acidic) and the worms try to escape. A handful or two of garden lime, or some crushed eggshells, will help to keep them happy.
If you get little flies in your wormery, try wrapping your kitchen waste in newspaper before you add it, or place a layer of newspaper over the surface of the waste in the wormery.
Make sure there's enough moisture in the waste, and that the drainage is working properly and the water level in the bin is not rising. And keep your worms at a comfortable temperature; they can overheat in summer and freeze in winter, just like us!