You can start leveling up other classes the moment you unlock the armory system (which should be after the level 10 class quest for your first job as I recall).

Given the levels your mentioning, it sounds like you misunderstood something. To unlock White Mage, you need to take Arcanist to level 15, and Conjurer to level 30. WHEN you do them doesn't matter, and you can take both classes as high as you wish (though you should unlock White Mage before taking Conjurer past 30.)

To begin leveling Arcanist (or any other class, including crafters and gatherers) simply head to the guild hall for the new class and talk to the receptionist to do a short quest for your level 1 weapon to switch to it. Remember to gear it up and do your arcanist class quests as you level, it has its own progress entirely separate from conjurer. It's almost like leveling a new character, but faster.

Also, I'm a little confused about your hunting log comment. *WHICH* hunting log were you trying to do? As what class?