Start young, visit often, and make every visit as enjoyable as possible. Socialize your kitten to being handled.
Store your cat carrier in a room where your cat likes to relax. Keep the carrier door open and put a comfy blanket or towel inside (bonus points for something that smells familiar, like your old fleece sweatshirt). Leave treats near the carrier door, then later inside for your cat to discover.
When it’s time to visit the vet, bring an old bath towel to cover most of the carrier and block the view of other pets in the lobby. Make sure there aren’t barking dogs present before coming inside.
Set the carrier on a table or chair (cats often feel safer in a perching position) and don’t let dogs approach to sniff.
Select a carrier whose top half comes off easily so that the cat won’t have to be “dumped” or dragged from it…nervous cats can often be examined without ever leaving the bottom of this type of carrier!
Use a calm, quiet voice and slow, reassuring petting.