Here's a simple but less-than-completely-documented way to keep Python package management under control:
1. Don't install anything globally. Don't pip install --user, definitely don't sudo pip install.
2. For each project you want to work on, create a venv. Yes, there are tools for this, but the base venv tool is totally fine. (venv should be included in your Python, but a few distributors like Debian put it in a separate package - install it from them if needed.) Use python3 -m venv ~/some/directory to create a venv. From here on out
3. As a first step, upgrade pip: ~/some/directory/bin/pip install -U pip.
4. Install things with ~/some/directory/bin/pip install.
5. Run Python with ~/some/directory/bin/python.
Slightly advanced move: make a requirements.txt file (you can use .../bin/pip freeze as a starting point) and use .../pip install -r requirements.txt. That way, if you get any sort of package resolution error, you can just delete your venv and make a new one. (Downloads are cached, so this isn't super annoying to do.)
A "project" can either be actual Python development, or just a place to install some Python programs and run them out of the resulting bin/.
(Edit: Yes, I know about activate, see the replies below for why I don't recommend it. With these rules, you get to say "Never ever type pip, only .../bin/pip", which is a good safety measure.)