Ever since we started having baby goats, we saw the need to milk our goat, either because she had too much milk, which caused her pain, or because we had to tube feed the kids.
To start milking, you must prevent the milk from flowing back by securing the top of the teat with your thumb and forefinger. After this, squeeze the milk out by applying pressure first with your middle finger, then with your ring finger and, finally, with your pinky finger, until the teat is empty. Then release the pressure to let the milk flow back to the teat and start over. Repeat these steps until both teats are empty.
Be careful not to pull on the teats, as this would only hurt the goat and would not make the milk flow any faster.
Here is a video of how to do it: