What should be the values of GOPATH and GOROOT?

GOPATH is discussed in the cmd/go documentation:

The GOPATH environment variable lists places to look for Go code. On
Unix, the value is a colon-separated string. On Windows, the value is
a semicolon-separated string. On Plan 9, the value is a list.

GOPATH must be set to get, build and install packages outside the
standard Go tree.

GOROOT is discussed in the installation instructions:

The Go binary distributions assume they will be installed in
/usr/local/go (or c:\Go under Windows), but it is possible to install
the Go tools to a different location. In this case you must set the
GOROOT environment variable to point to the directory in which it was
installed.

For example, if you installed Go to your home directory you should add
the following commands to $HOME/.profile:

export GOROOT=$HOME/go
export PATH=$PATH:$GOROOT/bin

Note: GOROOT must be set only when installing to a custom location.

(updated version of Chris Bunch’s answer.)