Since you are completely new, you aren't likely to be hiking long distances. Often, you can find a decent shoe on sale at a store like Sports Authority or Big 5. They aren't the best for a thru-hike, but you will likely get about 500 miles or so of wear out of them. They can always graduate to cutting grass later.

A hiking sole will get a better grip than a running shoe on slopes. I haven't used trail runners, so I have no opinion on those.

In my opinion, socks are as important to comfort as shoes. The two brands I recommend are Smartwool and Thorlos. One of them should be available in the box stores. They are made from wicking wool and will keep your feet dry from sweat. However, they can get soggy if you walk in the water. I'd suggest getting the socks before the shoes so you can try on the shoes with the socks you will be wearing.

Some people have great success with a thin synthetic sock-liner. If you are not susceptible to blisters, this can work out well as these socks dry quicker than wool. You will likely need a 1/2 size smaller shoe for these socks, though.

One thing to look for is the design of the shoe in the back. It is better if it looks like this ^^ (rounded.) Then the shoe won't be as likely to wear a blister on the back of your foot.

If you have any foot problems which may require an insert, I'd suggest going to a Red Wing store if you have one near you. They have a platform you can stand on and a computer will analyze the pressures on your feet.