For cats, travelling is made much easier if they are comfortable with being in their carrier. Time spent getting your cat used to their carrier is time very well spent. The key to this is not just getting the carrier out when you need to take your cat somewhere. Try feeding your cat in there from time to time and put their favourite blanket or toy in. Make sure you give yourself plenty of time to get them into their carrier as this will help ease both of your stress levels.
Dogs who aren't used to travelling may start to show signs of anxiety and vomiting as soon as they realise that they are about to go in the car. It is a good idea to introduce puppies to the car gradually from an early age, so they associate it with good experiences, such as walks.