Easiest way is to have another person helping you -possible?
Milk out other goat and put milk into jug and jug into warm water. I normally have the kid/lamb over my right leg and dangling down between my legs (gravity helps the tube go down). If you prefer to stand have the right leg up on a rung of a hurdle or milking stand or similar, again have kid dangling down between legs (if you are right-handed that is). Then just hold kid's body behind head with left hand, take bottom end of tube and slowly push into mouth from the side nearest to your body and down over the tongue. Just keep pushing down gently - the kid will swallow. I know everybody is frightened to get it down the lungs, but I honestly never have managed to do that... the tube should fit almost all the way down, and yes even with the tube the right way down the kid may splutter/gurgle a bit. (If you were to go down the lungs you would not be able to get it n till the end of the tube, as lungs much higher up in the body) Once down, preferably get other person to slowly put milk into the syringe bit. I never use the plunger on, actually have thrown it away. It will go down quite quickly. Once all you want to get down her is down, grip the tube bit with both fingers (to pinch it closed) and quickly in one swift movement take it out. If you do a male kid/lamb he will pee on your leg while you are doing it - guaranteed. I don't wash the tube between kids, but rinse with hot water between meals so to speak.
Don't be afraid - it becomes a routine task really quickly. Hope this helps.