Should I do every LVL 1-15 quest around a starting city?

Hi guys!

I am a completionist by nature. I like doing every single quest I can find. But there is something that worries me in this game.
I have started in Ul’dah and currently doing quests in Central Thanalan. I am level 8, and don’t seem to have any problems leveling up – there are plenty of quests and other XP sources in this particular area.
On the other hand I see that Western Thanalan is pretty much a similar starting area – the quests there and lvl 1-15 quests, etc. Instinctively I want to go to Western Thanalan and do all 1-8 level quests before I completely outlevel them, but shouldn’t I be saving them for other classes? Like if I would like to level my toon as Pugilist at some point, and all the quests in Western Thanalan are done, I will have nothing to do? Or am I getting all this completely wrong?

And what about hunting log? There are monsters there that live in Western Thanalan – should I just go and hunt them, ignoring everything else in the area?

Depends on how fast you want to level other classes later. Keep in mind when you go back to level other classes, you won't have the main story quests anymore. Side quests you save now will be the only quests you have for XP. That might not be a problem, depending on what content you enjoy and what leveling pace you're comfortable with. Personally, I haven't had a problem leveling other classes through 20ish with just FATEs, hunting logs, and instances, and the occasional side quest (I've avoided side quests for the most part on my main).

As far as the hunting log, you can go either way, really. Your quests will eventually send you there, so you can wait until then, or you can venture over now and hunt some things down, explore a little for the sake of it.

I try to save some of the side quests for alternate class low leveling progression. Mind you I try but then if I see a quest marker I usually have to do it. I guess like you I am a completionist also. The hunting log cannot be done by another class. Say you have thaumaturge and pugilist. The Pugilist cannot so the hunting log for the thaumaturge. The Grand company huting log I think can be done by any class. But Yes do the hunting log. That will give you a lot of XP.