How do I tell Homebrew to stop running brew update every time I want to install something?

Just prefix your install command with HOMEBREW_NO_AUTO_UPDATE=1, like this:

HOMEBREW_NO_AUTO_UPDATE=1 brew install somepackage

Source: brew manpage

I just modified /usr/local/bin/brew to add HOMEBREW_NO_AUTO_UPDATE=1 (according to @D Schlachter answer) at the start of the file

I personally find pinning/unpinning formulas more useful. For example you install a tool depends on 100 libraries which you don't use.

brew deps someprogram | xargs brew pin

Then you can check your pinned formulas anytime. If you have any problems with the version just unpin.

Not recommended to everyone, to pin all the formulas and manage updates manually:

brew list | xargs brew pin