As others said, get running shoes not cross trainers or other weird shoes.
If you've got a funky arch, try to get shoes that match up to your arch. A good shoe store/running store should be able to help you with that.
If you're looking for good shoes that won't break the bank, look at New Balance or Asics. They're both good running shoes and you can get a good pair for $50-70, way cheaper if you watch the Sunday ads for sales. I go through a lot of running shoes (outdoor, all weather running on concrete) so price is important to me since I replace them often. I was always a New Balance buyer, but my last two pairs I've switched to Asics because they were a bit lighter.
If I were you, I'd concentrate on longer, lower speed runs. Try to work your way up to 3 or 4 miles per session (3-4 times a week), at whatever pace you can handle. Speed will come later. Wetrat's probably got lots of good tips - my training regimen isn't based on science as much as stuff I've picked up through sports and police/fire/military training.
Running will be hard on your joints and muscles, especially if you're carrying some extra weight and your muscles aren't used to it. Ellipticals are a good idea for alleviating that, but elliptical machines keep you cooped up in a gym - plus they're just kinda ghey.