How can I use Homebrew to install both Python 2 and 3 on Mac?

I would use pyenv You can install it:

$ brew install pyenv

To enable pyenv in your Bash shell, you need to run:

$ eval "$(pyenv init -)"

To do this automatically for Bash upon startup, add that line to your ~/.bash_profile. ¹

Usage:

Once you have installed pyenv and activated it, you can install different versions of python and choose which one you can use. Example:

$ pyenv install 2.7.5

You can check the versions you have installed with:

$ pyenv versions

And you can switch between python versions with the command:

$ pyenv global 3.3.1

Also you can set a python version for the current directory with:

$ pyenv local 3.5.2

You can check by running python --version:

$ python --version
Python 3.5.2

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¹ _{Homebrew used to instruct you to do this upon installation of pyenv, but the message was removed. For Zsh and other shells, the precise steps may be different.}