The top list is from the main trainer, and to get one skill, you must get all the ones above it. The second list is skills that you can get by themselves, without any prerequisites. The third list is quest only. Change "Fire" to any other class in the link to get the analogous lists for other classes. Some details are missing for level 38 and 48 quest spells, but the rest are there.
Well my young Wizard and correct me someone if I am mistaken but as far as I know your secondary school spells are the ones you see on the list. your primary school will have spells not on the list but your seconday school pretty much follows the list.