Basically there are 2 reasons for doing a quest for any guild. First, to get the stated reward and second, to increase your reputation within that guild. 28 days after joining that guild (and every 28 days thereafter) you become eligible to be elevated in rank depending on how high your reputation and (appropriate) skills are. An increase in rank means an increase in perks from that guild. (IMO the perks from the fighters guild are lousy).
I know of no quests that you HAVE to do for a guild. If you don't want to do a quest just say no, there are no repercussions for refusing a quest. (DO NOT, however, refuse to do a Main Quest quest).
In the Fighters Guild, for the first month, the only quests you'll get are those small, random errands like you described. IIRC there is only one dungeon crawling quest you might receive, going after a werebeast. But even that is just a more complicated errand.