What are the reasons for choosing one over the other? If you have specific features and traits in mind then a purebred cat may be what you're looking for. If not, then you'll surely be able to find the type of cat you want by searching for a mixed breed. Some purebreds that are bred for specific traits may have some accompanying medical issues. With Persians for example, their snub noses can cause breathing difficulties. Mixed breeds can certainly have medical issues but they haven't been bred with man-made modifications that could create secondary health concerns.
Keep in mind that if you're thinking of a purebred it will cost you more money. Depending on the particular breed, it can cost hundreds and sometimes thousands.
Longhaired cats are gorgeous but they also require more maintenance. Most longhaired cats have coats that mat so daily grooming is required. With some longhair cats you may also have to do occasional bathing. If you're interested in the Spinx cat then frequent bathing will be needed. With a longhaired cat, be prepared to acclimate the kitty to daily grooming. Shorthaired cats benefit from brushing as well but they don't require the same amount of grooming.
Once neutered or spayed, the sex of the cat won't matter. Left unneutered, you will almost definitely end up dealing with a male cat who sprays. Unspayed females become vocal and make it their mission to get out of the house to find a mate. Spaying or neutering your cat will help you avoid lots of behavior problems and will reduce the chances of cancer in the cat later on.
Some people believe that if you already have a cat and are considering getting a second one that it should be of the opposite sex. I have never followed that. I believe it's the cat personality that is most important when choosing a companion for your current cat.
With an adult cat you can, for the most part, see the personality that has already developed so if you're looking for very specific personality traits then an adult would be a better choice. If you choose a kitten then you do have a good opportunity to shape the personality but you'll also have a greater time commitment. Kittens need more supervision and training as they're just learning about their physical abilities and don't yet know the rules of the house. Kittens will also need more frequent veterinary visits initially to get their scheduled vaccinations and dewormings. With a kitten you'll also have the upcoming needed neuter or spay surgery. If you choose an adult cat from the shelter then vaccinations will have been done and you stand an excellent chance of selecting one that has already been spayed or neutered.
A cat will be a member of the family and will hopefully be a part of your life for many, many years. Read about cat care and training so you'll be ready right from the beginning to start your kitten or adult cat off on the right foot. Too many people choose cats instead of dogs because they're under the false impression that cats are low maintenance and don't require any training. That attitude will result in an unhappy life for the cat and frustration for you. Take time to learn what your new cat needs to live a happy, healthy life.