To install Python 3.8 on Ubuntu version 23

Open your terminal and run these commands:

Install build dependencies

sudo apt-get update

sudo apt-get install -y build-essential libssl-dev zlib1g-dev libbz2-dev \
                        libreadline-dev libsqlite3-dev wget curl llvm \
                        libncurses5-dev libncursesw5-dev xz-utils tk-dev \
                        libffi-dev liblzma-dev python3-openssl git

Download and extract Python 3.8 source code

mkdir ~/python38
cd ~/python38
wget https://www.python.org/ftp/python/3.8.16/Python-3.8.16.tgz
tar -xf Python-3.8.16.tgz

cd Python-3.8.16

Configure the build

./configure --enable-optimizations

Compile the source code

make -j$(nproc)

Install Python

sudo make install

Verify the installation

python3.8 --version

To create a virtual environment specify the Python version.

Example: python3.8 -m venv venv

This post is community wiki so as to not claim the credit of @Kulfy's comment. This procedure worked on Ubuntu 18.04.

DON'T EVER CHANGE DEFAULT PYTHON!!! It may cause your system to break and some applications won't even run. It's far far far better to invoke python3.8 using python3.8 command

When installing python3.8, do the following

$ sudo apt-get install python3.8 python3.8-dev python3.8-distutils python3.8-venv

For most people this will be acceptable as they will be using a virtual environment for development. Construct a virtual environment and activate it as you usually would. This will leave you in a terminal where python resolves to python3.8:

$ python3.8 -m venv dev3.8/
$ source dev3.8/bin/activate
(dev3.8) $ which python
...dev3.8/bin/python
(dev3.8) $ python --version
Python 3.8.0

Neglecting to install python3.8-venv will result in an unhelpful error, that suggests you should install python-venv which resolves to python3.6-venv:

$ python3.8 -m venv dev3.8/
The virtual environment was not created successfully because ensurepip is not
available.  On Debian/Ubuntu systems, you need to install the python3-venv
package using the following command.

    apt-get install python3-venv

You may need to use sudo with that command.  After installing the python3-venv
package, recreate your virtual environment.

Failing command: ... (trimmed for formatting)

That article is outdated, most likely written at a time where Python 3.8 was not yet available on the stable Ubuntu repositories.

You can install the latest Python available with just apt install python3 (3.9 at the moment).
If you really want 3.8, I suggest you take a look at the deadsnakes PPA. It contains an archive of all Python packages for Ubuntu, and also newer versions that haven't hit Ubuntu stable yet (like 3.10).

You can add this PPA and install Python 3.8 with just three commands:

Copyadd-apt-repository ppa:deadsnakes/ppa
apt update
apt install python3.8

Regarding your other two bullet points:

• Yesn't. The thing is that, by default, /usr/bin is included in your PATH environment variable, which what enables you to just type python ... instead of providing the absolute path for the binary. It is also expected by some scripts, I suppose.
• You could, and you could also symlink it. The reason why it wasn't put there already is because the article used altinstall, which prevents exactly that. It is way more preferable that you install stuff through your package manager, however.

While we can use pip directly as a Python module (the recommended way):

python -m pip --version

This is how I installed it (so it can be called directly):
Firstly, make sure that command pip is available and it isn't being used by pip for Python 2.7

sudo apt remove python-pip

Now if you write pip in the Terminal, you'll get that nothing is installed there:

pip --version

Output:

Command 'pip' not found, but can be installed with:

sudo apt install python-pip

Install python3.8 and setup up correct version on python command using update-alternatives (as done in the question).

Make sure, you have python3-pip installed:
(This won't work without python3-pip. Although this will install pip 9.0.1 from python 3.6, we'll need it.)

sudo apt install python3-pip

This will install pip 9.0.1 as pip3:

pip3 --version

Output:

pip 9.0.1 from /usr/lib/python3/dist-packages (python 3.6)

Now, to install pip for Python 3.8, I used pip by calling it as a python module (ironic!):

python -m pip install pip

Output:

Collecting pip
  Downloading https://files.pythonhosted.org/packages/36/74/38c2410d688ac7b48afa07d413674afc1f903c1c1f854de51dc8eb2367a5/pip-20.2-py2.py3-none-any.whl (1.5MB)
  100% |████████████████████████████████| 1.5MB 288kB/s
Installing collected packages: pip
Successfully installed pip-20.2

It looks like, when I called pip (which was installed for Python 3.6, BTW) as a module of Python 3.8, and installed pip, it actually worked.

Now, make sure your ~/.local/bin directory is set in PATH environment variable:
Open ~/.bashrc using your favourite editor (if you're using zsh, replace .bashrc with .zshrc)

nano ~/.bashrc

And paste the following at the end of the file

# set PATH so it includes user's private bin if it exists
if [ -d "$HOME/.local/bin" ] ; then
    PATH="$HOME/.local/bin:$PATH"
fi

Finally, source your .bashrc (or restart the Terminal window):

source ~/.bashrc

Now if you try running pip directly it'll give you the correct version:

pip --version

Output:

pip 20.2 from /home/qumber/.local/lib/python3.8/site-packages/pip (python 3.8)

Sweet!