The reason running shoes are not great for rock climbing is because they tend to fit a little loose (since your foot swells when you run), they don't have hard soles, and to be durable enough for heavy road use, the rubber is not very sticky. Climbing shoes are typically have sticky rubber, are form fitting, and have fairly rigid soles so you can stick to smaller features. If you have any other shoes that fix one of the problems I mentioned about running shoes, they'll be better. Often times hiking boots are much easier to climb in than running shoes.
If you want a shoe that you can climb in but still use as an "everyday" shoe, you could buy a climbing approach shoe. It's basically a specially designed trail shoe that can be used for some climbing/scrambling. I'll occasionally use these when I'm climbing v2 boulder problems or 5.8s routes outside instead of changing into my climbing shoes. Unfortunately, approach shoes can cost almost as much and sometimes more than shoes specifically designed for climbing. Although they provide a nice compromise between climbing shoes and shoes you can use elsewhere, they aren't a replacement for climbing shoes. It's just hard to beat the confidence you gain when using a climbing specific shoe, especially when you're starting out.
My advice would be rent climbing shoes if you can; they give you the best experience and idea of what climbing is like. Plus, you won't be destroying your running shoes while your getting into the sport.