I agree with Laidback on his comments. To add my own:
It's all balance and compromise.
Depending on your personal preferences, you'll probably want a good lift in the bass department and a bit better quality of sound. But how you achieve this depends primarily on budget.
The biggest change you can make is to simply add a sub and nothing else. It's the bass frequencies that get absorbed the most in the car. Most of the power in music is at the lower frequencies, so if you boost these back up, your music will come alive.
But, as Laidback said, you really need to junk that headunit.
£200-300 is enough to make a decent system change, but you'll have to go secondhand.
A good quality new headunit will be around £200 on it's own.
Don't replace the door speakers, they can be fed from the amp and should be 'high passed' so that they only receive treble and mid range frequencies. This way, they will be able to produce a higher output.
You can leave them un-filtered, but you'll have to be careful you don't kill them.
So, a decent 4 channel amp. New, around £200-250. Secondhand, half this or less. Go for decent brands. Cheapos will not last and may badly colour the signal.
New headunit, around £200 (Pioneer or Alpine). Not really worth getting secondhand unless you know it's history.
Sub and box. Vary variable. New around £50-100. Half, secondhand.
Wiring. Around £30-50.
The headunit is the most expensive part, but should be as this is the first point in the signal path. From this point on, the quality can only deteriorate. So it's needs to start off good.
The amp is fixed in the boot. It is earthed with a thick cable to the chassis. It is powered from a separate, high current cable connected direct to the battery (via an appropriate fuse).
The RCA leads from the headunit are fed to the boot and connected to the input of the amp.
The outputs from two channels of the amp are fed via speaker cable to behind the headunit where they are connected to the existing front speaker wires.
The other two amp outputs are 'bridged' together to power the sub which is fixed in the boot.
The rear door speakers can continue to be driven from the headunit, but faded down a bit.
That's the basics. There's more detail obviously and it's a bit fiddly, but very well worth it. We can guide you through it.
There's plenty of research material on the web if you search.
www.bcae1.com Very thorough from start to finish on all aspects of car Hi Fi.
www.talkaudio.co.uk The ICE site. Very qualified and experienced people here.
If you want some piccies of how I did mine, I'll post them and bore everyone rigid, once again!