Fresh Air Junkie mentions the Five Ten Coyote Lace-Up as a great shoe for the beginner who wants to learn to climb. They tell you how to get started and what equipment you will need.
In regards to your climbing shoes: "Your selection must also be a good all-around shoe, as there are many specialty shoes on the market. But how can you possibly choose a specialty shoe when you haven’t even climbed a specialty route yet? If you stick with the sport, you will eventually buy a second pair of more technical shoes for this application. But leave that for the future. Right now you need equipment that will work for what is happening at your current level.... Five-Ten's Coyote Lace Up is an example of an entry-level shoe that can advance with you as you progress. The shoe features leather uppers with Stealth C4 rubber."
Check out their article "Rock Climbing: A Tip to Getting Started" to learn more.