Well you can start here if you are familiar with hardware Newegg.com - Computer Parts, PC Components, Laptop Computers, Digital Cameras and more!
and know how to put it all together.
You could get a barebones systems and then added the extra needed parts.
For $400 if you look hard enough, you can get just want you are asking for. Find a nice but cheapo case and put the $$ in the Processor, Ram, and Power Supply and a decent Hard Drive.
Buying a new computer wont make the internet faster but it may seem like it with a new system. Your ISP speed is the same speed no matter what computer you are one, the same with the downloading.
You could even save some $$ if you reuse say the case on your old system and possibly other parts.
I'm sure someone here can put ya one together on NewEgg when they see this.