I am about 7 hours in, and I have bought a few new skills for my Stryder toon, I have assigned them to my 3 slots, but I can't for the life of me use any of them.

For instance, I'm using a mouse and keyboard and I have 3 types of bow skills, but there is no indication of how to use any of them, there is one that fires 3 arrows at once, but that never happens, and one that fires a volley to a specified location, and the one is supposed to be assigned to E, but pressing E doesn't at all work.

What on earth am I doing wrong?

"Skills Set" actions allow you using "active" skills (there are also core skill that you use with Light or Heavy attack).
There are two skills sets: Primary Skills Set for primary weapons like staves, daggers and swords and Secondary Skill Set for shields and bows. Also, staves may use Secondary Skills Set in case of blue Vocations.

Anyway, press respective Skills Set button to display the list of 3 skills you have equipped along with action keys (Light Attack, heavy Attack or Action AKA the Use key) used to launch them.

Simply observe what is displayed on the on-screen button prompt - note how some skills have charging bars AKA gauge.
Also, read the details on how to use individial skills - press "Switch Explanation" when inside any menu that lists skills
https://i.imgur.com/X8F7DAL.png

Skill Sets are not toggle actions, you have to keep holding the pressed button to see the prompt for longer than a split second. The skill use sequence would be: you hold the respective Skills Set button, press the respective action key and immediatley let go of Skill Set button. If the skill is chargeable you just hold the action key. You can even launch skills by pressing both mentioned buttons simulatneously.
It's a very simple and reliable skills system used in many slashers, I'm sure you can handle it.

BTW Personally, I don't even have the prompt enabled. 3+3 skills are easy to memorize and I can use the audio/visual cues to tell if the skill is charged.