The type of climbing you intend to do will inform the type of shoe you want to buy; one shoe can be ideal for indoor sport routes and completely impractical for a multi-pitch climb.
For indoor climbing and bouldering, a shoe with exceptionally soft rubber on the sole and a downturned toe is perfect. The soft rubber grips small holds extraordinarily well and there is less of a chance of tearing up the sole than there is on natural rock. The downturned toe allows the climber to hook into jibs and pockets more easily, especially on overhangs that are so ubiquitous in the bouldering caves that most gyms offer. The drawback is that the shoe is even more uncomfortable than the average climbing shoe and the rubber is more likely to wear over time.
For outdoor, top-roped climbing or free soloing, a slingshot shoe with a harder grade of rubber is the most advantageous. The fit is more comfortable and the shoe can be worn for long periods of time. The design forces the foot forward in the shoe, ensuring that the toes have the most power possible on all holds. A thicker rubber takes away some of the stability on small holds, but it will last longer and hold up better against wear.
For the climber who intends to do a lot of heel hooks or crack climbing, the full heel and full toe rubber shoes are the best. Rather than just having climbing rubber on the bottom of the shoe, it wraps all the way around the toes or all the way up the heel. This maximizes the contact with the rock thereby creating the friction to hang on, but it sacrifices breathability which means the foot can start to slide around within the shoe when it starts to sweat.
One of the most important features of a climbing shoe the way in which it stays securely on your foot. The options are lace-up shoes or Velcro/elastic slipper shoes, each with its own perks and drawbacks.
Lace-up shoes are generally more durable and will continue to fit snuggly even after the leather stretches and wears over time. On the other hand, the shoes are less comfortable, heavier and allow less flexibility in the sole.
Elastic slippers tend to be more comfortable and are easier to put on and take off, but just as the leather stretches over time, so does the elastic; the foot is more prone to slide around. This problem is alleviated with a Velcro strap to tighten the shoe, but even the Velcro can wear over time.
A climbing shoe shouldn't fit like a normal shoe. A climber should generally buy a shoe that is at least a whole size smaller to ensure that the foot stays snug in the shoe and that the toes come together at the tip, allowing the most toe contact on small holds. When wearing a climbing shoe, the toes should feel crammed in the end. As uncomfortable as it may be, it will be worth it on the rock.
Climbing shoes require a lot of care and depending on the type of shoe you choose, it may require more maintenance. No shoes should ever be exposed to extreme heat. Leaving shoes in a hot car can cause the barge cement holding the rubber to the leather to separate and lose its adhesiveness. Climbing shoes should also never be worn when you're not climbing. Downturned toes can lose their shape, and the soft rubber on the bottom will wear down. Climbing shoes should be put on and taken off at the base of the climb.
Shoe creators are currently working to ease the hassle of taking climbing shoes off and on constantly when moving from one climb to another. They have created an "approach shoe" which is essentially the love child of a tryst between a climbing shoe and a hiking boot. The outcome is not ideal for either, but it works moderately well for both. The shoe contains more padding and support than a climbing shoe, but still has climbing rubber along the bottom. Shoe designers are still making advances in this type of product and in the future, the approach shoes could potentially make hiking boots and climbing shoes obsolete