Ok, you want the CD&T vaccine, not the antitoxin.
Water based personal lubricant.
Some old towels
Short fingernails
A bottle (I actually usually use just regular baby bottles) even if you aren't planning on bottle feeding.
A packet of ReSorb electrolyte solution. You can use any homemade stuff to give a doe after a normal kidding, but if she is really exhausted from a bad kidding, or if a young kid scours, you'll want the good stuff.
I have raised goat for going on a decade, and only twice have I ever actually caught a healthy doe in the act.
Nope, I just look out and there she is with babies. I've even slept in the barn, went in to grab food and a quick shower and there she goes!!
On the incredibly rare occasion I've had to help DON'T PULL.
Just don't.
First, gently ease a very clean, well lubricated (and seriously, lubricant is cheap, get it everywhere, on everything, use a LOT) hand in and make sure that whatever legs and noses there are are all attached to the same kid. Gently ease the baby into the diving position, with one hoof further ahead of the other. DON'T PULL. Gently, gently, gently help the tissue stretch around the baby.
If the hips seem caught, DON'T PULL. Hold the front legs, one well ahead of the other, down towards the Doe's hocks. Hold, DON'T PULL. Just don't let the baby slip back between contractions, and make sure that one side (leg) is ahead of the other, so that shoulders and hips aren't coming out square, but at an angle.
If you really, really think you need to pull, get a vet. Because pulling is bad, and if you rip the cervix, you've just killed your goat, and it's a slow, ugly death.
I have had to use the hard core part of that knowledge all of 3 times over the years, twice with other people's goats. Mostly, you don't even get to see it.
In early years, on the advice of an experienced friend, I would lube up and reach in to check and make sure she was done an had passed the placenta, but experience ha taught me the less you interfere, the better.