Here is our Red Doe giving birth to twins (03-08)
The clear fluid-filled bag is the sack around the kid to be born. You should soon see two front feet and the nose of the kid emerging from the birth canal. If all is correct you will see the two front legs come first, with the kid's head rested on top. The next step in the birth process is the appearance of the head. Most of the time a thin membrane will still cover the kid. This membrane should break as the kid is born. However, you may want to be nearby in the event that the kid needs assistance. If the birth is progressing, allow nature to take its course. Only assist if necessary. A water bag with red fluid appears after the first kid is born. 
Remember that a young doe who is giving birth to her first kids will take more time to complete the birth process than an older, more experienced doe. This first-time doe took 40 minutes between the first and second kid.
As more of the kid appears outside of the doe, allow the doe to continue pushing until the kid is completely born. When the doe stands up, the umbilical cord will break on its own. Do not cut the cord as this is likely to cause excessive bleeding. Allow the cord to tear on its own. Once the kid is born, check to make sure that it is breathing. Wipe the head and nose off well to make it easier for the kid to breath. If the kid is not breathing, try inserting a piece of straw a short way into a nostril to encourage the kid to sneeze. You may also need to lift the kid up by the rear legs and vigorously rub its sides. Don't rub all the "gunk" off of the kid, let the Doe lick it off, this will ensure that she will claim it and you won't have to bottle feed it 6 times a day for 3 months.
But in cold weather it is a good idea to dry off the ears and tail as best as possible. This helps to prevent freezing. Allow the mother to lick the kid to clean off the rest. She should be "talking" to her kid now that it is born. You can use a towel to dry off the kid's head and nose. This will help them to breath easier. Also place the kids at the doe's head to encourage her to lick them off.
Here are both kids just seconds after being born
It is very important that you give them some bonding time (about 15 minutes). But then once the mother and kids are in the kidding pen you need to check her teats (by milking them just enough to make sure they are open and to check that the mother has milk). Then grab a piece of dental floss about 6" long and tie it about 1/2" down from the belly onto the umbilical cord very tightly. Then cut the umbilical cord with sissors to where it is about an inch long and dip it in iodine to prevent any navel infections. I use iodine that I have poured in a film container (washed and disinfected with alcohol), some people pour it in a baby food jar, anything that has a tight lid will work. Once the kid can stand I put him on her teat and make sure that he gets a good drink before leaving them alone for the night.
Here they are a few hours later
