If you're going to be running seriously, ditch the cross-trainers and get honest running shoes. I won't recommend any brands because it's all subjective, but any shoe that is in a specialty running shop is going to be a quality shoe. Go there and ask for help getting fitted for the right shoe. It might take a long time; that's ok. You want to find a shoe that's right for your foot and gait (they will help you with that) and then you have to go by feel. Try on as many different pairs as you can, and jog a bit in each and buy whichever feels best.
Also, yes, rotating different pairs of shoes does help. I have 3-5 different shoes that I wear throughout the week; the time off helps them dry and they say it helps the foam to rebound; I'm not so sure that's ever been proven but it can't hurt, right? Also, one subtle thing to note is that rotating two different shoes (like completely different models) can be beneficial as different shoes make your foot work in slightly different ways. So instead of working the same muscles in the same exact motions day in and day out, switching between two similar but different shoes can actually reduce your chance of injury. Again, I'm not sure exactly how much of an effect this has, but it's something to think about.