I'm a gamer, just to get that out there right away, and it's time for some upgrades. Right now I have:

CPU: AMD 3400+
Mobo:Asus A8R - MX/S
Video card: X1650 Pro
RAM: 1 GB DDR
HDD: Western Digital 160 GB
PSU: Bastec 300W

So, upgrade or build a new one? I've been thinking of building a new one but if I can get away with an upgrade I will. I'll take either suggestion, and recommended upgrade/build parts would be nice.

It is all a question of budget. I am presuming it is socket 939, first thing to do is upgrade memory to 2GB, next would be new GPU like hd3850 and looks like PSU as well. A x2 3800+ or 4200+ is another possible upgrade.
Total cost for all those upgrades would be between 350 and 400USD so it would be up to you to chose between upgrade and new build. With all the upgrades you would still have a good mainstream gamer.

I would recommend starting from scratch with a new Intel build. Depending on budget i recommend:

Single 8800GT (£170)
Core 2 Duo E6750 (£100ish i think)
A decent-ish P35 board (£65)
Antec Sonata III case (Comes with 500 Watt True Power II PSU, which is great) (£60)

This should give you a great upgrade with potential to upgrade to a Quad Core in the future.

If that's a bit too steep i recommend:
HD3850/3870 (£100/£140)
AMD X2 5000+ (Not the black edition) (£55)
Cheap motherboard (£40-50)
with the same case

That should be a fair bit cheaper and give you a MUCH better gaming experience than your current system could ever dream of.

Sorry if you're in the US, i'm just not familiar with US prices so can only give you them in sterling.

I'd recommend a new system too just for the fact that things have gone so far since your current rig. But if you do plan on upgrading, I would first and foremost upgrade to a better PSU. That is one of the most important parts and is many times an afterthought. Make sure it's a well-known brand with enough wattage and amperage to power your final spec comfortably. I don't think what you have now will be up to the task of running a new video card, dual-core processor, and double the RAM, especially if you're spending any considerable amount of time gaming.