$300 is stage 1 (yeah, I know and you will only ever buy one pair of headphones
). If you try and spread $300 around for the whole system, you will be disappointed. As
earthpeople said, look at replacing the front stage speakers.
Get a HU that is more future-proof, ie something that can play or connect to MP3 players, iPod etc. Also one that has enough power to comfortably drive your front stock upgrades and rears when you get them, otherwise wait and save some more.
The rear is important for sound reinforcement. If and when you can run a separate amp, unless it's a big multi-channel, use it to take over running the front speakers and/or sub. Leave the HU to run some rear 6x9s if you need it. The rationale is that you are using the best amp to run the best part of the system.
To put it in context in my car the front stock 6.5 have been replaced by
Hertz speakers. The rear parcel shelf has stealth fitted
Focal 6x9s with a 12"
JL Audio Sub Wedge in the boot. The front and sub are powered by a
Soundstream amp and the 6x9s by by the
Alpine CDA-9855R
Linked to my Ipod Photo 4G by the Alpine KCA-420i.
The beauty of such a setup, is that you don't push it to enjoy it. I can listen to classical to drum and bass as quietly or as bangin' as I please. The point is I didn't start out with all of this. I built it up bit by bit. Getting the best components for each section I could afford. Pretty much as I intend to put together my headphone system.
What sort of car is also important. If it's small you probably won't need the rear 6x9s, just change out the fronts add an amp and a sub. If the car's a keeper, custom the doors and drop some 8" and put some custom tweeters up top to give you some decent stereo imaging.
But what ever you do, as you also wanted, have fun.