
Hi. I think your vet has muddled you! You're quite right, you need to train your puppy, little and often, as much as you can early on. It's important that lessons are short and sweet, and always finish on a success! Collars and leads aren't unkind, and are an essential. A small light collar is fine for a small pup.
I suggest you buy a copy of "The Perfect Puppy" by Gwen Bailey - it's chock-full of excellent training and rearing advice.
:)