What should be the values of GOPATH and GOROOT?

Here is a my simple setup:

directory for go related things: ~/programming/go
directory for go compiler/tools: ~/programming/go/go-1.4
directory for go software      : ~/programming/go/packages

GOROOT, GOPATH, PATH are set as following:

export GOROOT=/home/user/programming/go/go-1.4
export GOPATH=/home/user/programming/go/packages
export PATH=$PATH:$GOROOT/bin:$GOPATH/bin

So, in short:

GOROOT is for compiler/tools that comes from go installation.
GOPATH is for your own go projects / 3rd party libraries (downloaded with "go get").

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.)