I have:

AMD Athlon 64 3400+
1GB RAM
and my rather poor 9600 vanilla.

Would an upgrade to, say, a 6600GT be a substantial boost in performance? What is the cheapest place in the UK to buy graphics cards?

I'd say you'd notice a pretty big difference in upgrading that card, almost to the point of being similar to buying a new computer as far as gaming performance is concerned.

You'll more than double your framerates, and in some games you'll triple.

Have a look here: http://graphics.tomshardware.com/graphic/20050705/vga-charts-pcie-07.html

(look at the 9600 pro card, which is actually better than the one you have now, and how it compares to the 6600gt).

Just be sure your power supply is enough to run your new video card.

What is your motherboard, by the way?

Is your AGP slot 8X?

My Motherboard is a "ASUSTek Salmon GL6E"

My AGP port is 8X.

Yep. Looking good!

Like I said, your only worry is whether or not your current PSU will be enough to power that new card.

I'd say a MINIMUM of a 380w psu (anything less and your system may become unstable), but recommend you get a 450w. But most importantly, make sure you have at least 24amps on your 12v rail. You can find all this info on the sticker of your power supply (inside the case).

If you find you'll need a new PSU, look at companies like Antec or Enermax. Very reliable PSUs.

I don't know the cheapest place to buy in the uk, but I've heard that overclockers.co.uk is a good place to buy.

Here's an agp 6600gt, the top one here.
http://www.overclockers.co.uk/acatalog/6600AGP_Series.html

I'd recommend a psu of 350w or more(a 300w might run it). I've ran my gaming rig, 3000+ overclocked, 1gb ram, 6600gt, etc, with a 350w psu with 22a on 12v, and also with a 420w psu with only 18a on the 12v.
I checked here.
http://www.extreme.outervision.com/index.jsp
and the OP's rig would be using less than 300w when I put in the specs I know, although you shouldn't be using the psu's max power continuously.