Would compiling programs from source speed up my operating system?

Somewhere I read that I should remove my default programs in Ubuntu and then build them from source to speed up my system. Would this work?

No, and it wouldn't be worth the time it would take you to do it, not to mention the fact that you wouldn't get security updates directly from Ubuntu anymore.

The only time I would build myself is if you need a version of the software not in any repository, then consider downloading the source. Download the dependencies (development libraries etc) from Ubuntu repositories and build when you need to update.

I don't want to seem disrespectful, as I include myself in this, but the package maintainers on the whole will produce a far better build than you will. Unless your build options enable some sort of hardware acceleration, you won't see a lot of difference.

I've compiled complete systems from scratch using linux from scratch and gentoo and found I only felt performance improvements on really lower powered and specific hardware, such as the Playstation 2 and a 1ghz Atom cpu on a mini-ITX board. Bear in mind that I was required to do this on these systems as normal compilation flags would not work on the defaults.