Hi Karen,
You've received lots of great advice already, and my book also address this topic, but here's a bit more info so you can get started ASAP. If the doe only has a single kid, you really should have started milking her right away for maximum milk production. Obviously you can't go back in time, but I'm just telling you this so that you're not disappointed by her production, and you'll know that next time she can produce more.
Since she only has one kid, she probably has a little milk in her udder all the time, so as others have said, just put her on the milk stand and try to milk her. First thing in the morning is usually best in terms of getting more milk because kids don't nurse a lot overnight. I wouldn't separate her from the kid for more than a couple of hours initially because it's more difficult to get a let down from a goat that has a very full udder, especially when you're new. If you can't get a let down, you can let the kid nurse for a few seconds to get her started. When we have a doe nursing only one kid, we put her on the milkstand twice a day, both morning and evening to milk her out. She should be able to produce
enough milk for two kids, so I say that I'm the other kid in that situation.
In a very small herd, the least stressful way to separate the kid is to put it in a dog crate right there next to mom, so they are not really separated, but the kid can't nurse.
You didn't say whether the kid was a buck or a doe, but if it's a doe, you don't have to wean her. In fact, it would be quite difficult because you don't have another goat to keep her company without leaving the mama alone. Goats are herd animals and should be kept together in groups of at least two, although more is better. If this is a buckling, I'd suggest just castrating him so he can stay with mama.
Hope this helps!