You could ask 10 triathletes that question and get 10 different answers.
The best thing to do is to go into a store, find a selection of shoes, try them on and literally run around the store in them (many stores will even let you run around the block in them).
If you have "fussy feet" or think you have special needs, go to a store that specializes in running shoes. They can asses your stride and footfall to get your something that will be good for your foot.
Running shoes are usually rated for flexibility and stability, you also have weight and comfort which is totally subjective. There are a myriad of factors that can play into which shoe you get. I'm seasoned runner, I look for shoes that are light, super flexible which usually come with minimal stability by virtue of being light and flexible, but I am a strong runner so I don't require the stability that others might.
I love running shoe shopping, it's like starting a new relationship! I always take my time and try on many and try to make a good decision because I know I will be spending a lot of quality time with them.
I usually have two pairs of running shoes, an old pair that is comfy and familiar to my feet and a new pair that I know has better support and better for racing. The new pair is en route to being comfy and familiar. I wear my racing shoes infrequently, just for racing once they are broken in. After their two marathon life, I retire them as my gym shoes which I wear more often but never for long periods.
I'd also suggest getting "Yankz" http://www.powertri.com/run/running-accessories/yankz-shoe-laces.aspx
they are elastic shoe laces that you can set to a comfy tightness and still slip over your feet, no more fussing with laces. Plus, they are a god send when you are transitioning in a tri.