How does a WMG level up if its not in a group and about how long would it take to get to level 18?
Despite the limited strength it should be very possible to level up a White Mage solo till around level 10 I'm guessing.

Also, can anyone way the pros and cons of being a WMG in terms of gameplay?
Well, for starters you don't have to rest quite as long as other people thanks to cure. And as a WM you'll also have no trouble finding a party later on. It's kinda dangerous to be without one. The cons of a white mage include the fact that if the creature your party is fighting in a suddenly starts to target you, you've got a good chance of dieing right there, and if you cast a high level healing spell or use the WM super ability in battle, there's a really good chance that you'll be the next target of the creature.

I was thinking about being a monk or a white mage and I want to know wih one would be a better starting job.
Might be easier for you if you start with a Monk so if you start leveling up as a White Mage later on you can put Monk as your subjob to give your White Mage some of his abilitys and stats. (only if you leveled up Monk past level...18? I forgot the subjob requirements)