No job specific type traits will transfer to another job really.

Say you level your Black Mage and then decide to level Warrior... you will probably go Warrior/Monk for a while and will basically get nothing from the Black Mage leveling you've done. You will get some mild benefits if Black Mage is your subjob though. If you go White Mage/Black Mage you will get whatever abilities your subjob has to the half level mark of your main job.

Bio is a level 10 Black Mage spell (iirc) so when your White Mage hits level 20 if they are subbing a 10 (or higher) Black Mage they will be able to use Bio and all of the Black Mage's other abilities to that point.

Additionally any 'skills' that you learn transfer between all of your jobs if they can use that ability. A few examples...

If your sword skilled up while leveling Warrior from say 0-50... if you go back and play Thief and use sword you will not have to gain any levels in sword. At each level there is a skill cap so for Thief at level 3 or so you would have 12 max sword skill. If your sword is already 50 overall then your Thief will reap the benefits of having a level 12 (already maxed) sword. Additionally when your Thief hits 4 and your skill cap jumps up to 16 or so it will once again already be capped and you will not have to reskill it for each job.

However, all of that sword skill will have no impact on a job that cannot use sword such as Black Mage or White Mage. Even if you find the rare swords that a Black Mage can use Black Mage will have no sword skill so the sword will be essentially useless other than any stats it may bear that affect other attributes of your play.



Sorry that the answer to your question was so long and complicated.