How to set Python3 as default in MacOS Monterey (Macbook Air M1)?

If you are on Mac M1

Steps

1. $ brew install python
2. $ sudo ln -s -f /opt/homebrew/bin/python3 /usr/local/bin/python
3. Restart your terminal.
4. $ python --version

If you are on Intel Mac

Steps

1. $ brew install python
2. $ ln -s -f /usr/local/bin/python3.<your version> /usr/local/bin/python
3. Close Terminal and test version 
4. $ python --version

You can use symbolic link: ln -s -f /usr/local/bin/python3 /usr/local/bin/python

From the information provided it is not possible to say if /usr/bin/python is already a symlink. You can check all the python versions in that directory and see if it is a symlink or not.

ls -l /usr/bin/python*
ls -l /usr/local/bin/python*

If python is an existing symlink

In the case that python is a symlink, you should be able to change the symlink to link to python3 as already suggested here, -s for soft, -f for force to overwrite an existing symlink.

ln -sf /usr/bin/python3 /usr/local/bin/python

If the operation is not permitted you will need to create it as root. If this is good solution I do not know, so use my answer with precaution.

sudo -i
ln -sf /usr/bin/python3 /usr/local/bin/python

Check if the right version is recognised

where python
python -V

From MacOS Monterey 12.3

From MacOS Monterey Version 12.3, python2 is no longer preinstalled, and is shipped with Python 3.8.9. Therefore python is not in use and a symlink can be used without concerns regarding python2.

python -V
zsh: command not found: python
python2 -V
zsh: command not found: python2
python3 -V
Python 3.8.9