Yes there are quests on the mainland, and they're quite varied (killing mobs, crafting items, harvesting, delivering post to npc's that are obscurely positioned, etc etc)
You'll need to do some to get your fame up unless you want to stay neutral / neutral.
I've also heard that there are quests that give you increases to your stats but I haven't looked for those yet.
I've found the fame quests a bit of a drag personally but I need to get them done, and they don't really take very long so not a major issue really.
Basically, aside from getting the required fame, if you want to do them you can, if you don't want to you don't have to.
One important thing trial players need to know
Re: One important thing trial players need to know
I don't think "quests" is a particularly appropriate name for the missions available on the mainland. They're tasks that you accept in the hope of earning dappers or fame upon completion. One does not lead onto another, and they may all be repeated as many times as you like. Then again, searching unaided for either a wandering npc in Heretics Hovel or a forage source of Select Yellow Spotted Dried Butterfly Legs is about as epic a mission as you will find ... ever 
On the other hand some of these tasks also happen to be parts of Encyclopedia rites. The rites are quests of a sort as they have a progression starting with a rite initiator such as a Barman, are not repeatable, and reward you at each stage with a part of the Lore. Upon completion they reward you with a unique piece of knowledge - an ability or craft plan or bonus available in no other way - and also a sizeable piece of Fame. As it stands now, I think there are only 12 fully functioning rites? Each step in a rite requires you to perform one of the tasks from one of the usual mission givers so Fame of one sort or another becomes a requirement so that they will even offer you the job in the first place.
On the other hand some of these tasks also happen to be parts of Encyclopedia rites. The rites are quests of a sort as they have a progression starting with a rite initiator such as a Barman, are not repeatable, and reward you at each stage with a part of the Lore. Upon completion they reward you with a unique piece of knowledge - an ability or craft plan or bonus available in no other way - and also a sizeable piece of Fame. As it stands now, I think there are only 12 fully functioning rites? Each step in a rite requires you to perform one of the tasks from one of the usual mission givers so Fame of one sort or another becomes a requirement so that they will even offer you the job in the first place.