Types of shoes
Flat/neutral shoes are designed for comfort and longer routes. If you are climbing short, overhung routes, look into a more aggressive downturned shape. Laces can help you get a more precise fit but take longer to put on and off. Velcro shoes are easier to take on and off, but may not get as precise a fit or be as good for crack climbing.
Sizing
Your toes should just touch the ends of the shoe in a well-fitted climbing shoe. They should be snug but not painful. The type of climbing you do will also inform how you want your shoes to fit. If you are doing long multi-pitch climbs, you may choose to go for a more comfortable, flat/neutral, and less-snug shoe. If you are climbing for short periods of time (e.g. bouldering or hard sport routes), you may choose to go for a tighter fit for better performance.