If you want to install Python 3 using Homebrew:

brew install python3

Output:

==> Downloading http://python.org/ftp/python/3.3.0/Python-3.3.0.tar.bz2
Already downloaded: /Library/Caches/Homebrew/python3-3.3.0.tar.bz2
==> ./configure --prefix=/usr/local/Cellar/python3/3.3.0 --enable-ipv6 --datarootdir=/usr/local/Cell
==> make
==> make install PYTHONAPPSDIR=/usr/local/Cellar/python3/3.3.0
==> make frameworkinstallextras PYTHONAPPSDIR=/usr/local/Cellar/python3/3.3.0/share/python3
==> Downloading https://pypi.python.org/packages/source/d/distribute/distribute-0.6.35.tar.gz
Already downloaded: /Library/Caches/Homebrew/distribute-0.6.35.tar.gz
==> /usr/local/Cellar/python3/3.3.0/bin/python3.3 -s setup.py install --force --verbose --install-li
==> Downloading https://pypi.python.org/packages/source/p/pip/pip-1.3.1.tar.gz
Already downloaded: /Library/Caches/Homebrew/pip-1.3.1.tar.gz
==> /usr/local/Cellar/python3/3.3.0/bin/python3.3 -s setup.py install --force --verbose --install-li
==> Caveats
Homebrew's Python3 framework
  /usr/local/Cellar/python3/3.3.0/Frameworks/Python.framework

Distribute and Pip have been installed. To update them
  pip3 install --upgrade distribute
  pip3 install --upgrade pip

To symlink "Idle 3" and the "Python Launcher 3" to ~/Applications
  `brew linkapps`

You can install Python packages with
  `pip3 install <your_favorite_package>`

They will install into the site-package directory
 /usr/local/lib/python3.3/site-packages
Executable python scripts will be put in:
 /usr/local/share/python3
so you may want to put "/usr/local/share/python3" in your PATH, too.

    See: https://github.com/mxcl/homebrew/wiki/Homebrew-and-Python

Once installed update your system PATH variable, add the next line to the ~/.bash_profile file:

export PATH=/usr/local/bin:/usr/local/sbin:~/bin:$PATH

And then:

source ~/.bash_profile

Now launch Python:

python3

Output:

Python 3.3.0 (default, Mar 26 2013, 10:01:40)
[GCC 4.2.1 Compatible Apple LLVM 4.2 (clang-425.0.27)] on darwin
Type "help", "copyright", "credits" or "license" for more information.

You can check the python3 path:

which python3

Output:

/usr/local/bin/python3

How to install Python packages for brew installed Pythons?

Some Python packages can be installed via Brew, e.g.:

brew install python-setuptools

In the Terminal, type:

brew link python

For Apple silicon machines, the paths are slightly different. After running brew install python, you must ensure your ~/.zshrc file uses the correct Homebrew paths:

# Homebrew
eval "$(/opt/homebrew/bin/brew shellenv)"

# Homebrew: Python
export PATH="/opt/homebrew/opt/python/libexec/bin:$PATH"

Results:

% which python
/opt/homebrew/opt/python/libexec/bin/python

% python --version
Python 3.9.9

% which pip
/opt/homebrew/opt/python/libexec/bin/pip

% pip -V
pip 21.3.1 from /opt/homebrew/lib/python3.9/site-packages/pip (python 3.9)

Brew's Python interpreter changed its location again and I had to update my ~/.zshrc file to:

export PATH="$(brew --prefix)/opt/python@3/libexec/bin:$PATH"

There isn't a .../opt/python/... anymore which is annoying, because now I have to specify v3 explicitly, but at least better than having to specify the minor version like in the comments.