How to best speed up the client?

Any tips that'll improve framerate and reduce clientside lag noticeably?

buy a new computer

There are a some ways. One of them being Optifine. Sadly that one does cause problems with rendering on some mods. So it's not an option.

Increasing memory usage might help a bit.
Turn off as much as possible in terms of other programs hogging the computers resources.

I can't really say much since I've always aimed to having a beast of a computer. Just to avoid ever having problems with graphics and performance.