I have those drenching syringes from Jeffer- they are really easier to use. But I did use regular syringes for awhile- I put it the side of the mouth and put it as far back on the tongue as I could and squirted a little bit then wait for a swallow then more, etc.
Oh- I forgot to mention- I don't hold the head up- I try to hold it level. I think if it's tilted up it tends to cause more problems swallowing. Then I hold the chin level with a finger on the throat- when I feel the swallow I give more.
Goats are easy for me I think because when I was young, drenching was the way to get many medicines into horses. The vet would leave you with this powder or that and you would have to get it into the horse by putting in a bottle, lifting the horses head so it was level than pouring the drench in little by little. I can remember trying to hold a giant horse head with my arm straight up in the air until I couldn't hold any more, then having the drench all come back on the top of my head as their head came down if they hadn't swallowed. Goats are easy- really easy. Oh so easy.