I'm not saying there is a wrong way but just wanted to share how we do it. It's not a "easy" process but we take extra steps that make it harder on us though it ensures clean milk.
First lets start with the goat
Luckily goats stay pretty clean anyway and this may be over kill. I like the feel of a freshly shaved goat. I use the Whal hair trimmers you can find at Walmart. I like the adjustable kind where i can use no guard but extend the lever about half way and trim about halfway of the under belly to the "bum" of the goat. All of this is a big help in both keeping hair out of the buckets and easier to clean that udder.
I have a cure for those hairy stray teats that like to divert the milk stream to your pants.
They are simple nose trimmers. I use the green one above and you can get one for $4 after shipping on ebay. They are gentle and don't leave cuts or nicks on the teats.
Now to the goat kit. It's a simple tool box that I can bring inside during the winter so it don't freeze. I start with a clean rag and a spray bottle of soapy water and give her a good scrub. Then I pull a couple of unscented wet wipes and put a good amount of Purell on it and give a second scrubbing concentrating on the teats. This is also good to help sterilize your hands. Next is a "strip" bucket I leave out to get a good 4 to 5 first squeezes to get that first "dirty" stream out.
Now I'm ready to milk. A few modifications I made to our milk bucket though.
We put 2 frozen Avian water bottles in the bucket. The goats milk comes out sterile and its how you handle it keeps it sterile. The colder you can get it the fastest is the best and you can put those water bottles in the dish washer after you strain the milk.
I also went to Home Depot and bought that steal looking non rusting screen that I cut and form fitted to the top of the bucket to catch any hairs or bugs that may try to get in there. (dishwasher safe also) I top the bucket off with a heavy dinner plate when I set the milk bucket aside. When I'm done milking, the bucket goes straight in the freezer while the goats finish their meal at the stanchions. I take the milk out and put it in the fridge just before it starts to freeze.
After milking you want to sterilize the teats as the last step. Some people use Fight Bac in a can but I been using Purell for years.
Above can be your worst enemy. Just like us, goats don't like flies landing on their legs. Most goats don't stomp high enough but if you get a sensitive goat, a little fly spray on the back of their legs helps. Just keep that milk bucket covered and a good distance. Now for butthole goats that give you a hard time like the one pictured at the top it took me years to figure her out. Nothing makes you more mad then when a goat "kicks the bucket." She would put her hove in the bucket and even lay down in it making all that work useless. After years I finally figured her out. All it took was my head pressed in her side. When I would feel her twitch I would give her a nudge knocking her off balance and when she would lean into my head the other direction I would let up and this method worked great. And if all else fails, thump them like a read headed step child :p no offence to gingers :p
Now the last steps. Filter the milk and we put it in glass Ball jars (4 equals a gallon) Then we to back out and kick the goats :p It depends on what type of goats you have how long it will last before it gets a goaty taste. A good supplement that grows fast is Zucchini and Squash. They love it. Just don't feed moldy hay. While it wont hurt them it will give them the Hershey squirts :p
When kidding season comes, be sure to shave that belly, udder, bum, and tail or you will have a long time sticky mess.