Yeah sounds like you did it fine
The only two other issues I've come across are - air drawing up into the syringe when you pull back the plunger to check it's not in a blood vessel...I was sure I hadn't stuck the needle through the other side and couldn't work out why on earth there would be air under the skin, until the vet pointed out to me that it means that the needle is too loosely attached to the syringe and air is creeping in at the joint. If this happens, withdraw the needle, push the needle hard onto the syringe and reset it without air in it before trying again.
The only other problem I've ever had is that the needle gets blocked and you can't push it in, this can particularly happen with some antibiotics that are a bit 'thicker' than others, and especially if they've been lurking for several days waiting to be used. I now keep a spare needle at home so I can swap the needle over myself without needing to faff about going back to the vets for a fresh syringeful.
And my final piece of advice - I use a 500g bisto gravy granules jar to store the needles - it's a perfect size for keeping them in without risk of accidentally pressing the plunger!
Hope all is ok