Will it speed up web surfing? Probably not. The speed at which you access the web is generally not determined by your computer, but by the speed of connection that your ISP provides AND the speed at which the particular server you are connecting to, can deliver the data.
Will it speed up multi tasking? Very likely yes. Almost all new computers now are multi core and many are 64 bit systems. If you have 64 bit capable programs, you will certainly see a better performance.
Will I be able to transfer memory? It's highly unlikely your memory in a ten year old computer is compatible with a new machine.
What should I be looking for? Not trying to be flippant, but how long is a piece of string? It primarily depends on your budget. Even an entry level machine now provides perfectly acceptable performance. But if I were you, assuming you wish to use Windows, I'd wait for Windows 7.
Better still (though I accept unlikely), you could like me use Linux and discover what true performance really is!