Ruthven is one week old today. He's experienced some ups and downs, but he seems to be doing quite well.
We were initially feeding him Sav-A-Kid milk replacer, which is basically powdered milk with various nutrients added. I had trouble getting him to nurse from a bottle, but he took to drinking from a bowl almost instantly, which has made our lives much easier.
He did alright with the milk replacer for a couple days, but Tuesday he came down with a case of milk scours. This can be a very serious condition leading to dehydration, so we took drastic steps. We stopped his milk altogether and gave him nothing by water and electrolytes (table salt, potassium salt, baking soda and corn syrup) for a day to clear out his system. When the diarrhea stopped, we resumed the milk replacer. Unfortunately, the scours returned, so I went out and bought a bottle of whole (cows) milk and some heavy cream to enrich it slightly (whole cows milk is 3.25% fat, goat milk is around 4%). He seems to be doing better with the real milk.
Ruthven is an amazing ball of energy. A human baby one week old is just a helpless lump, barely able to do anything but eat, poop, cry and sleep. Ruthven, on the other hand, follows us around the house, runs about the yard, tries to play with the dogs, and jumps about on the furniture.
When we first brought him into the house, we put him in a plastic storage bin that we dubbed the "Goatquarium" to sleep during the night. Unfortunately, that only lasted a few days, as he quickly gained the strength to leap out, despite the walls being above his head. Now we are keeping him in an old dog crate. It has plenty of room for him, as it was designed for a fully grown Great Dane. We've lined it with old carpet pieces and towels and added a salt lick, some hay to sample, a cardboard box to hide in, and a bowl of fresh water. But that is only at night, during the day he's either being cuddled by my SO or myself, or running about the house under careful supervision.

The Goatquarium
I'm beginning to wonder if a goat can be housebroken. Every time we take him outside, he immediately pees on the grass. He pees a lot in the house too. I'm amazed that one little goat can produce so much urine. That's a good sign, though, as the urine is colorless and odorless, so he's not getting too dehydrated.
Luckily he doesn't have much of a scent. He gets a daily bath (mostly because of his messy poops), so any scent he'd normally acquire is neutralized.

Goat huffing. Kids, just say no to goats.
Ruthven gets along well with the two dogs. Our female Great Dane seems to have some slight mothering instinct towards him, and is protective of him. Our male Lab initially exhibited some hunting behaviors (stalking, etc), but whenever he made even the slightest aggressive move, he got a beat down from the Dane and a verbal reprimand from us. He's learned his lessen now and accepts the goat as one of the pack.
Our ram doesn't show the slightest interest in Ruthven, while our ewe shows a very strong maternal instinct, despite her advancing age. If she could still produce milk, she'd have probably raised him herself. Our remaining nanny goat seems rather aloof to the kid, maybe because he's not hers. And his father, perhaps sensing a future rival, occasionally headbutts him. We're eventually going to have to sell one of the males, and I'll be damned if it's Ruthven.
Our biggest concern however, is the donkey. He's showing very clear aggression towards the baby goat, even whirling around and trying to kick him. I'm not sure if he just hates babies, or if he thinks the goat is another dog who needs to be put in his place, since he comes out of the house just like the dogs. If the donkey is out on the lawn, we are especially vigilant around him.
Ruthven is just the absolute sweetest thing. He loves to be wrapped up in a nice warm towel and just cuddled. He's particularly fond of my SO and will often cry if he gets out of sight for long. Ruthven has a very healthy set of lungs.

This is one happy, spoiled goat.
Ruthven is nibbling on
everything now, so we have to be extra vigilant when he's playing. The house is pretty well dog-proofed, but a baby goat is so much smaller than a Lab and can get into so much more.
Last night he was in the kitchen watching his mama (my SO) preparing his milk, so I laid down on the ground. He started climbing all over me, then started exuberantly licking my face. He'll probably start sampling solid food soon, if he hasn't already. That should help his digestion settle down even further.

A young Gene Simmons?