We put substrate, food, bedding, veg and a dog biscuit in the carrier, as well as a toy if the pet has one that we think will be safe during travel. Then as it can be pretty windy here we have a shirt that we cover the carrier with during the short walk to the car, seatbelt them in the car and off we go.
Most of our pets have never minded attending the vets at all - two of the hams have had stressful visits and gone off the vets after that, but generally ours take it all in their stride
If a hamster has had a stressful visit then on future visits we try to spend as much time handling them as we can in the consultation room (e.g when the vet is fetching medication or explaining something to us) and its really helped get the hamster to relax when there again - Fwirl didn't mind, then had a stressful visit and didn't want to go back, but now she enjoys going there because as far as she's concerned its just a drive in the car and a lot of playtime