Getting the right running shoes is one of the most important things you can do as a runner. In fact, having shoes that don't fit properly is one of the leading causes of leg injuries among runners. Ignore brand names, colors, fancy advertising - none of that matters. The first thing you need to do is find a running store (ie, a store that focuses specifically on runners & running - NOT a big chain sporting goods store) and tell the clerk that you are in the market for your first pair of running shoes and need a fitting. They should measure your foot in several different directions, watch you walk and run both barefoot and in shoes, and have you try on several different types of shoe. (If they don't do this, go somewhere else.) Don't worry about what the shoes look like - go with the ones that feel the best on your feet.
As for the price, you can expect to spend between $100 & $150 on a quality pair. I know it seems like a lot, but wearing the wrong shoes can end up costing much more in doctor and physical therapy bills. Shoes are really the only major expense when it comes to running, though, and having the right pair is incredibly important in terms of injury prevention, so don't skimp, on matter how much you may be tempted. Once you've been fitted and know what type & brand of shoe works the best for you, you can look for sales & bargains when it's time for your next pair.
Finally, be sure wear your shoes ONLY for running and no other activities (the motion involved in non-running activities can cause running shoes to break down faster), and to track how many miles you run in your shoes. Most pairs last around 300 - 500 miles, depending on the runner; after that, the midsoles wear out (even though the outer part of the shoe may look fine) and it's time for another pair.
Best of luck!