Your cat's vet is an incredibly important person in both your lives so choose wisely. In fact, your vet is as important to your cat as your doctor is to you and your family. Please, don't wait until a Vet emergency to start looking. In fact, I would suggest finding a vet before or as soon as you bring your new cat or kitten home. Speak to the people where you got your kitten from. If you live in a different area, do some research and ask your friends or the local pet shop for recommendations. And as vets deal with many different types of animals, asking around will help you find one that is particularly good with domestic animals.
Vets unfortunately can't ask their patients what the problem is. Therefore it is up to you to be able to explain the symptoms or ask questions. I have had friends who have had trouble with this as their vet was overseas trained and had English as a second language. He was an excellent vet but there were language difficulties which can be distressing in an emergency. Doing your due diligence when doing your research is incredibly important.
I suggest contacting different vets in your area. If you live in a suburban area, there could be vets in neighbouring suburbs which gives you a greater choice. Ask for a list of their fees for a general consultation, vaccinations, nutritional or other advice. Your vet may have a plan that encompasses routine visits for checkups or booster shots. Also find out the opening hours and if you need an appointment before bringing your cat in. Some surgeries run 'open hours' where you can turn up announced but there can then be a lengthy wait. My vet requires appointments but this means we are in and out fairly quickly. Ask them what the procedure is in an emergency. Can you just arrive? Do they have evening or weekend hours? Does the practice have its own hospital for after surgical care? I would also ask the vet about alternative treatments such as acupuncture, chiropractic, massage and homeopathy. The benefits of these complementary therapies can be substantial, especially in older cats.
Once you have your cat or kitten, schedule an early visit with your vet. This will allow him (or her) to do an initial exam to get to know your cat. The earlier he learns about your cat, the easier it will be for him to detect problems later. He can check for parasites such as fleas, ticks or worms. He will also listen to the kitten's heart and lungs to make sure there are no problems in that area. Most vets also check for discharges or discolouring around the eyes, ears, nose etc and make sure the teeth and gums look healthy. He will also probably check for infectious diseases such as feline panleukopenia or feline infectious anaemia. If by some chance your kitten or cat has an infectious disease, it can be caught early so it isn't passed on to other pets in your home or neighbourhood. If your cat or kitten needs vaccinations, this can be done at this visit. I suggest making a further appointment for having your cat spayed or neutered as soon as it is old enough, unless you intend breeding.
Having confidence in your vet will certainly ease your mind as you know that your precious cat or kitten will be well looked after should the need arise. And if you have taken your cat to the vet from an early age, he will have a record of inoculations or other problems. It is exactly the same as your doctor having a complete medical history of your children. Your vet will probably look after your cat throughout its life so make sure you are happy with him and his methods. The freedom from worry will make life much less stressful for you and your cat.