What step am I missing installing drush with Composer?
Well, actually you installed it locally. You can find it in your project's directory under vendor/drush/drush place (maybe, it depends where the composer put it). You can call only from that place because it's not in your global path. If you want to use this method, you need to call it with an absolute or relative route. Like if your composer installed to here:
On Windows:
D:/phptest/my_project/vendor/drush/drush
On Linux:
~/my_project/vendor/drush/drush
and you are in the D:/phptest/my_project (Windows) or ~/my_project (Linux) dir, in that case, you need to call it like:
On Windows:
vendor/drush/drush --version   OR  D:/phptest/my_project/vendor/drush/drush --version
On Linux:
vendor/drush/drush --version   OR  ~/my_project/vendor/drush/drush --version
and you will get something like:
Drush Version   :  8.1.3
Of course for this, you need a PHP in the global path. (so you need to be able to call PHP from anywhere.)
If you installed it globally with composer (I think you just installed it locally), in that case maybe you need to create a (soft) symlink one place which is globally available for you (listed in the %PATH% variable on windows or in the $PATH variable on Linux), or you need to add the drush's dir to your global path variable.
I suggest you, install it globally, like here is written: https://docs.drush.org/en/8.x/install/
# Browse to https://github.com/drush-ops/drush/releases and download the drush.phar attached to the latest 8.x release.

# Test your install.
php drush.phar core-status

# Rename to `drush` instead of `php drush.phar`. Destination can be anywhere on $PATH. 
chmod +x drush.phar
sudo mv drush.phar /usr/local/bin/drush

# Optional. Enrich the bash startup file with completion and aliases.
drush init

Then you can use it in any site's dir.
TIPP: if you're installing drush launcher (https://github.com/drush-ops/drush-launcher) globally, then you can have per-site base different drush installed. In this case, you don't need to have global drush installed.