I'm on level 10 but can't seem to access the third bonus skill so haven't passed any checkpoints. What am I doing wrong?
The third bonus skill is "Christmas" and won't be available until late December.
Your level has nothing to do with accessing bonus skills. To access bonus skills you have to complete the first six or so skill sets and then buy the bonus skill from the lingot store. Your level is based on XP (experience points) which you earn by completing lessons, but if you do every lesson once you will be able to buy the bonus skills at a relatively low level, whereas if you do every lesson five times you will be at a higher level when the bonus skills become available for you to purchase.
Passing a checkpoint has nothing to do with bonus skills. Some courses do not even have bonus skills. To pass a checkpoint you have to either complete all the lessons in all the skills leading up to the checkpoint, or take a test and get enough answers right to "test out" of the skill sets and pass the check point that way.
Your next lesson is lesson number 4 in the skill set "People". This means you have already passed two checkpoints and are close to passing the third.