Kropacius

Do you possibly have another machine or network connection you can try on?

Please delete your WTF, Cache and Interface folders.

Then try the game again.

Nashe, can you try rebooting your router again.

Worldchamp, please make a new thread.

Have you been able to connect at all recently?

You may have to turn hidden files and folders on in order to locate the Battle.net Cache:
http://windows.microsoft.com/en-US/windows7/Show-hidden-files

Please go to:
for XP: - C:\Documents and Settings\All Users\Application Data\Blizzard Entertainment\Battle.net
for Vista/7: - C:\ProgramData\Blizzard Entertainment\Battle.net
And delete the Cache folder in there

Please also power off your Modem/Router and any other network hardware for about 5 minutes then restart it.

My personal opinion would be to go ahead and get a new system if at all possible, but as others have said there's a fair bit of research to be done, Does your Mainboard have the right slots for the card you want, can your PSU handle its demands?

512 of RAM really isn't that much that would probably need to be supplemented or replaced, and again you need to check the mainboard compatibility with it.