The vet can be a very scary thing for many dogs. Especially if your dog does not have to make many trips to the vet, it can be quite frightening when and if he/she does. It could be very helpful to prepare your dog ahead of time to visit the vet. This way you, and your dog, will likely have a better experience.
One way to do this is to teach your dog to be okay with being touched. Your dog may be used to you petting him/her, but it may have a harder time when it comes to strangers approaching them with an outstretched hand. It may be helpful to expose them to this kind of gesture as much as possible so that they are used to it, and comfortable with it, by the time the vet comes along. You can even turn this into a touch trick. When your dog approaches your outstretched hand, reward them in some way. This will teach them that it is okay, and they will not be bothered when the vet does this.
Another thing you will want your dog to be familiar with is having its paws, ears, and mouth touched. These are common places for the vet to check out, so your dog will need to comfortable with it. Many dogs are not fans of these areas being touched. Your dog may quickly pull away its paws when you try to touch them, or they may scratch and bite if you attempt to see their teeth. You will want to teach your dog that these things are okay. You could try things like giving them a reward when they allow these areas to be examined, or you could try punishing when they do not listen. This will make it easier on both your dog and your vet.
Something else your dog will need to be okay with is being picked up, especially if you have a small dog. It is likely that your dog will be picked up, and some dogs do not particularly like this. Make sure your dog is used to being picked up so that it does not frighten them once it is coming from the vet. The other part of this step is that the vet will likely set your dog on an examination table. This may be very frightening as your dog is probably not on tables very often, if ever. You may want to practice this with your dog ahead of time. This does not necessarily need to be something that your dog does on a regular basis. You may not want your dog to constantly be on your table. However, doing this will help your dog to feel safer and more secure once it is on a table at the vet.
Your dog may still become skittish at the vet even with these steps, as it can be strange to be examined by someone your dog does not know. However, these ideas will help your dog to be more at ease so your dog can stay healthy.