Most veterinarians realize it is not easy to get your cat to the clinic when they're sick, let alone for regular health maintenance. It is always best to start before you need to get the cat to the vet. As those of you who are cat people already know, cats do not like change; they are most comfortable with the familiar. In most cases they are not familiar with the carrier, or only know that when it comes out bad things happen.
So, first let the cat know that the carrier is not an instrument of torture. The goal is for your cat to connect the carrier with good things so that he/she enters the carrier voluntarily. Make the carrier familiar at home by leaving it in a room where the cat spends time. In my household, when we were not home, the cats were confined in the basement. The carriers were used as Stripe, Banshee, and Biscuit's beds. Toys or catnip were placed in each one overnight. At first, the treats were removed quickly; over time the cats would go in and enjoy the treat and relax. One of my assistants, Dasha, uses her carrier to feed each cat individually. Each of her cats goes to "its" carrier for meals. This comfort level helps when it is time to go to the vet. This does take time, though how long depends on the cat. Sometimes you may need to give a cat treat just for the cat going near the carrier. It could take days, weeks, or (rarely) months to accustom the cat. Stay patient.
But what if the cat needs to go right now? Start by putting the carrier in a small room with few hiding spots. Bring the cat into the room. Move slowly and calmly; do not chase the cat, as this only frightens them. Encourage the cat with treats, toys, and/or gentle nudges to walk into the carrier. If the cat won't walk in and the carrier has a top opening, gently cradle and lower the cat into the carrier. You can also remove the top half of the carrier, put the cat in the bottom half and re-close the carrier. Using familiar bedding or a shirt with your scent also helps keep the cat calm. You can also use a synthetic feline facial pheromone (Feliway) sprayed inside the carrier 30 minutes before placing the cat in it. The pheromone helps calm the cat and make it less stressed.
In the above paragraph I mentioned a top-loading carrier. Feline Practitioners know that the best carriers are inexpensive hard-sided carriers with doors/openings on the top and front that can also be easily taken apart in the middle. This way the carrier can be seat-belted for travel, then when at the vet's office the cat can stay in the bottom of the carrier where it feels more secure for the exam. PetMate has several models that have these features.
I hope that with these recommendations, your cats can visit the vet with less stress, be better evaluated, and remain healthy longer.