How do I earn quest points?

By doing quests.

I've done a few and it hasn't shown any points earned

You don't get one every time, it's random.

You don't get quest points each time but if you do quests regularly they add up quickly. At the moment I have over 600 quest points. Since July I've must have earned over 1000 points. I'm always buying items, sometimes in bulk, from the quest shop.

I suggest adding the Quests widget to your sidebar, http://subeta.net/preferences.php?act=sidebar, and completing as many as you can daily. I personally only do the quests from the Saggitarius, Wizard, Library, Maleria, Quentin, Pete, Drunk Carl, and Cursed quests.