I'm a photographer at a university and do some (limited) teaching.
The problem with most point and shoot cameras is that you don't get as much manual control over the camera, and even with that limited control, you will find it difficult or impossible to get certain effects.
That said, I've had a few students use p+s cameras and they did learn something. (Usually learned that they wanted a DSLR. :)
Your local big box or Costco or whatever will have a very limited selection, but really any inexpensive DSLR from Nikon, Canon, Pentax, or even Sony will be fine. It will come with a "kit" zoom lens, which is also fine. It shouldn't be hard to keep this under $500. Don't feel like you need the latest model -- a previous generation will be fine, and usually much cheaper.
If you can hold out a couple of days, you can buy a used camera and lens kit from KEH.com. I've been using them for years. Buy something a couple of years old and get it for way less. They have a D3100 kit for $400 and a D5100 kit for $465. Call up and ask them for advice.