Most magic schools are easy to master. Follow this to master any or every school of magic early in the game.
1) Conjuration : the easiest of all , simply summon a ghost over and over , summoning is the easiest way to master conjuration.
2) Alteration : 2nd easiest to master , simply find a locked chest or door with an easy lock or harder. Cast "open very easy lock" over and over and your alteration will sky rocket.
3) Illusion : very easy to master , if you have an invisibility spell , cast it over and over , you can go about your missions , and cast invisibility in dungeons and caves etc , and your illusion will raise very quickly.
4) Mysticism : just as easy as illusion , cast a detect life spell over and over , you can go about your missions , and cast detect life in dungeons and caves etc , and your mistycism will raise very quickly.
5) Destruction : difficult to master unless you use this tip. Gain access to the arcane university , create a fire spell 3pts on self for 1sec. Cast the spell over and over , when you get low on health of magicka , rest for an hour , and repeat. Longer than the previous schools , but the reward is well worth it.
6) Restoration : the most difficult school to master. Healing yourself is a good way to raise your skill level , but try creating a restoration spell on TOUCH , approach people in the IC and heal them , also heal your followers , bodyguards , steeds and even the lil' skinned hound. Eventhough this will take a long time , it is still important to heal your followers , especially in the oblivion gates where npc's have a knack for getting themselves killed. You could also enter the arena and heal your opponent before he/she dies , effectively training yourself in sword play and armor as well.
All magic skills increase as the spell is succesful , regardless of its strenght , so weaker spells are more effective to raise your skill. Note: by mastering all your magic skills you might also have a harder time increasing intelligence and willpower later on in the game. A way to counter that is "perma-duping" how do you do that ?? stay tuned...