Listen... to be able to answer your specific questions, we'd have to at least know exactly which motherboard you have.
I can tell you this though... most likely, you'll need to pretty much do a complete build, as an old MB for an 1800+ processor isn't likely to accomodate late model processors.
You might be able to reuse the case, but plan on replacing everything else.. MB, processor, memory, power supply, video.
Yes AMD was great back in the day, but I think Intel has overtaken them with the Core-Duo processors, in both performance AND compatibility (and price!). My entry level Intel Core-Duo E6300 processor absolutely BLOWS AWAY my old 2200+ processor, and this is the 'low end' of the processor family, and an economy motherboard. Matched with an older 7900GS video card, I rarely see frame rates below 50 (1280X1024, 1024 hi-res textures, all settings maxed), and usually it's pegged out at 75.
From what I've read, the new Intel processors run very cool, and use very little power (compared to the P4 generation and AMD processors). I was a die hard AMD fan back in the day, but I don't plan on giving up my Intel any time soon! Personally I'd probably go with a faster dual core processor instead of a quad core (not much support for the quad yet), but others beg to differ.
Also you'll want to find a PCI-e video card.. might as well go DirectX10 as this is becoming the new standard. I recommend an NVIDIA 9600GT... the price is right and from what I understand it's an outstanding performer.
I wouldn't worry about DDR3 just yet... DDR2 is plenty fast enough for just about anything
Find a good quality 400watt or more power supply to go with it as these new video cards suck a lot of juice.
Now here come the replies with more specific recommendations...