When my Becker crapped out I decided to do a step-wise system upgrade starting with the head unit. My budget was similar to yours and my objective was "good enough to enjoy" without blowing the neighbors away, with the whole system being as stealth as possible. The installation I'd do myself.
For the head unit I wanted an ipod input, built-in HD Radio receiver, an amber display to blend in with the dash, and ability to connect to the stock amp. I got a slightly older version (this is 6 months ago) of this $170 JVC head unit from Crutchfield:
http://www.crutchfield.com/p_105KDHDR60/JVC-KD-HDR60.html?tp=5684 It came with the installation kit and when I had a few questions during the install their customer service was great. I'm very pleased with it. It's better in every way than the Becker.
Next I upgraded the dash speakers to Focal 100CAI 4" coaxials for $150 delivered from Woofersetc.com. I'm very happy with them as well. Lots of good choices out there. Whatever you choose be careful about depth, the absolute max you want to go is 55mm. There was someone on eBay selling a $20 adapter kit you'll need for any speaker upgrade, but I don't see it there now. There are tech tips with other options - do a search.
I got a pair of Focal mids for the doors but never bothered to install them because the factory mids aren't than bad and I have a factory sub box that provides more than enough bass, actually too much at times. If you don't have one, I recommend it because it's plug & play, sits out of the way on the rear deck, and not pricey if you have to buy one, $250-$300 is what I've seen them for.
In order to stay within your budget I highly recommend the step-wise approach I took. What you're happy with may be far less - or far more - than what other people will be happy with.