It's important to get the correct fit for YOU. Asics will sell a pair for most basic types of running, that will also correct for either overpronating or supinating. The also have multiple widths, arches, etc.
Saucony and brooks are both good shoes as well for running.
New balance is probably one of the best if not the best cross trainer.
Lastly, on the high end, I've heard great things about Newton Running shoes, but have never tried a pair myself.
The last thing you want to be running in is the Nike shox or similar type shoe.