To fortify Alchemy, Conjuration, and Mysticism, they will need to fortify Intelligence. This will increase the amount of Magicka the Hero has at their disposal). One way to help level-up in this area is to cast a Drain 'skill' spell (such as Drain: Block), because armor exp is determined by how many times the armor/shield is hit, therefore lower level with stronger armor ( Daedric is best) they get more experience, but costs loads of hammers to fix the armor. With Heavy/Light Armor/Block skills experience is determined on the amount of damage The Hero takes and endures, just make sure to have a decent heal spell that can be cast almost constantly. One option is to head into a goblin cave and find a pen of rats and let them attack you.

They will want to fortify the governing attribute of the skill they are training in, especially with magic skills. Skill Leveling Tips for The Elder Scrolls IV: Oblivion.