Ideally you’d want to introduce your cat to their harness as a kitten. Older cats can be trained too, but it might just take a little bit longer and therefore require a bit more patience. Please never force your cat outside. Put yourself in their paws – imagine being forced into the scary outdoors knowing you can’t run away. It is vital to allow your cat to feel safe and to trust you with the leash before heading outdoors for a proper exploration session.

Step 1: Get your cat used to wearing the harness indoors.
Allow your cat to get used to the sound of the velcro first. When they seem to not be bothered by the sound, place the harness on your cat without the leash attached. Give them a treat or two with the harness on then slowly take the harness off. Only give treats when the harness is on, do not give treats when the harness is off. Repeat this process while gradually increasing the amount of time your cat wears the harness.
 
Step 2: Get your cat used to walking on the leash (without tension) indoors.
After your cat is comfortable with the harness, attach the leash to the harness. Begin by allowing your cat to walk around as you follow with the leash loose. After a short time, remove the harness and leash and repeat this process for a few days until your cat is relaxed and freely walking.

Step 3: Get your cat used to leash tension indoors. 
While supervising, allow your cat to drag the leash behind him or her while freely moving around your home. This will allow your cat to feel and get comfortable with a little bit of leash tension. Always make sure to supervise this and never leave a leash or harness on an unsupervised cat.

Step 4: Walk your cat indoors.
Put on the harness and leash and follow your cat as in Step 2. After the cat has walked a bit, gently try to lead your cat in another direction. Use treats either dropped on the floor or held in your fingers to lure your cat and then reward for moving if you need to. Reward your cat with praise and treats if he walks on his own, too!

Step 5: Go Outside.
On a dry, temperate day, start by walking your cat to the door, opening the door, and encouraging him to go outside. To help with this, toss a treat one-foot outside the door to lure your cat. If your cat is frightened or hesitant, do not force him to go out. Instead, stop for the day and try again later.
Make sure to bring treats for your cat when walking outside. Keep the time spent outside to a few lovely minutes. When it comes to training, it is far better to end on a positive note than a negative one.

Source: The Anti-Cruelty Society

The following course has also been recommended for those who seek more guidance: Cat Harness Training