This is not a question we can answer for you. Climbing shoes are like ski boots - one brand will fit perfectly, while another will not fit at all. Personally, I have found that 5.10 fit my feet, whereas La Sportivas do not at all. For a beginner shoe, look for an all-around shoe, not a super aggressive sport shoe. Size them just a bit below comfort level, maybe half a size if synthetic. If you buy leather shoes, plan on them stretching quite a lot, while synthetics stretch very little.
Go to a local climbing shop, preferably a dedicated climbing shop and not REI. They will be able to fit you with something that will work well for you for quite a while.
Those are very aggressive shoes, as their name implies. To "project" a climb means to work or attempt a route, over and over and over again, wiring every move, until you can eventually "send" it. This is usually done sport climbing much more than trad climbing, and usually difficult climbs. If you are a beginner, which your posting makes me believe, this is the wrong shoe for you. I am guessing you and the people you are going to be climbing with are not psyched on spending all day making your way 15 feet up a 5.12 sport route.
I climb 100+ days a year between rock and ice, and travel rather extensively doing so. A good 5.10 shoe I would recommend is the Moccasym - http://www.campsaver.com/anasazi-moccasym-climbing-shoe This shoe is comfortable, and suitable for a wide variety of uses. It will not hold your climbing back, and it also will not limit you to a single type of route like the Project would.