OK. I just ordered a new head unit, and found a good deal on a set of four 6x8 speakers on eBay. I was looking for just 2 for the fronts, but came across a really good price on all four that was only a few bucks more than just 2.
I know that people in a lot of areas are not fans of Pioneer equipment, but I have had good luck with them in the past, and again, this isn't meant to be as incredible as all of your great setups. I did go a little over budget, but oh well.
Ended up with a Pioneer DEH-P6700MP head unit. It will work directly with the Pioneer CD Changer I already have. I also grabbed four Pioneer 6x8 speakers (two 3-way TS-A687OR and two 4-way TS-A688OR). I know, I was supposed to get Alpines, but I couldn't find a really good deal right now, and my job is taking me out of town without internet access for 2 weeks, and I would like to get things ordered and on there way so that I can get it all installed when I get back.
Maybe in a few months I will drop some money on an amp and sub, but for now, this setup will work and should be a good step up from factory.
Thanks for all of your help. The tip about replacing the head unit first really drove the price range on the speakers. I hope that I won't regret it later.