You can leave the babies with the mamas, provided that you're okay with getting the leftovers.
You might get a lot one milking but hardly anything the next, depending on when the kid nursed last.
Here is how I milk:
Before I go outside I gather all of my equipment. I use a five gallon stainless steel bucket (I milk a cow, you obviously don't need something so big for goats. My dh milks a goat and he uses a one-gallon milk tote from Caprine Goat Supply). I also get a container--an old coffee tin or yogurt container works well--and fill it with hot water. I add a tiny bit of dishsoap and sometimes grapefruit seed extract and tea tree oil. This is to clean the teats with. I add to that a clean dishrag. Many places, including Molly from Fias Co farms (google it, she has a ton of great information on goat husbandry), use paper towels but, again, I've got a cow and cloth works better for the hard rub-down I give her udder. You'll have to find out which works best for you.
Before heading outside I generally scald out my milk pail.
Outside I get my dairy ration and my back-up ration. The dairy ration is a mixture of grains and, as she eats this fast, my back-up is soaked alfalfa cubes. The grain helps her maintain a healthy weight as she puts so much energy into making milk for her kids (and for you). I pour the grain into her trough, sit on the bucket I just poured the grain out of, and clean her teats. (I should add that goats are much shorter than cows and when I milk goats I just sit on the ground, but you might find another arrangement that works better for you).
Some people also carry a small cup to squirt the first squirt of milk into. You don't save the first squirt of milk as it has bacteria. They look at this milk to make sure there are no streaks of blood or clumps in it (signs of mastitis). I generally don't do this but, again, you might want to. I just squirt the first couple of squirts on the ground, at the dog, et cetera.
Milking itself is pretty easy, once you get used to it. Hold the teat at the top, right near the udder. Hold it between your index finger and thumb. Using your other fingers squeeze down the udder (one at a time down to the pinkie). You don't actually pull the teat with goats, you are just squeezing it.
You'll find a rhythm of swish, swish
, back and forth, on the teats. You'll feel as she lets her milk down and you'll feel as she starts to empty. Her teats, which were full, will now start to be shrivelled. Massage her udder (some people kind of slap the udder)--you're mimicing the butting that the kid does to get the last of the milk to let down. Watch how her kids do it and you'll see what I mean. Generally it's very important to get all of the milk out--not only is milking a supply and demand operation but also leftover milk can cause mastitis but since her kid(s) will be with her it's probably less important.
When you're done take the milk inside (it's important to cool the milk as quickly as possible so don't do other chores while your milk waits) and strain it. I do this by pouring the milk through a stainless steel strainer covered with cheesecloth into glass jars. Then I put the jars into an ice water bath (we use large coolers specifically for this purpose, but again we're dealing with quite a bit of milk). When we had less milk we just did this in the sink. You want the milk to cool as quickly as possible to lessen the amount of time bacteria has to grow.
You can, of course, pasteurize the milk but we do not. There are instructions on the internet to do this if you'd like to, it can be done on the stove.
Wash and sterilize your milking equipment and put it away until next milking. I'm very fastidious about milking and do it between 8:30-9:00 every morning and evening. My dh is less punctual and just milks once in the morning hours and once in the evening hours.
We don't tie our animals to milk them but many people build or buy stanchions to hold their milking animals. For a goat it's a platform that they stand on (raising them high enough that you can sit comfortably on a chair to milk without breaking your back) with some sort of v-shaped thing that they put their head through. Their feed is on the other side of the 'v' and the 'v' holds their head in place while you milk. There are free instructions for building these online but, again, depending on the temperment of your goats you might not even need one. They are handy, as you can use them not only for milking but for hoof trimming and other medical needs, but those things can be done without one.