HOw do you increase your notable level (for quests?)

You have to research the Adventure techs.  Read the tech descriptions and some of them say that they open up quests. 

This doesn't answer the question.  I am not aware of an adventure tech that says "increases the notable level of your soverign".  But I HAVE seen adventure techs that create new quest locations with a higher notable level then my character currently has....

I have to agree that it is not as clear in the techs descriptions as it should be, as to which stat notable/adventure they raise but it is still how its done..

For the kingdoms there seem to be three path in the adventure tech research ... and although I don't remember the actual names, their order seems to constant. So let's just call them No.1-3

• No.1 - This seems to be the Heroes and Monster level branch. Researching it increases the level of spawning monster and champions for hire. If you don't like a dragon in your backyard beware of this branch.

• No.2 - Quests and goodie huts (or in game terms notable locations) are here. They rotate. Researching it the first time increases your quest level, the next time the notable location level, then again questing ... you get the point. The text and name of the respective tech mostly gives a hint on which of the one of the two levels will increase.

• No.3 - Secret resources. Not 100% sure to the effect myself. One at least reveals orchards and/or bee hives. The shard spawning during the game could also be related to this. As might the appearance of other resources. Odd things is, that everybody seems to profit from researching into this branch. Not just the one actually researching it.

I hope this is more or less correct and answers your question. As for the Empire tech tree? No idea. Haven't touched the dark side yet.

Rabenhoff

Notables are also called goodie huts and other things. It's basically anything that isn't a Quest or an Adventurer/Creature.

What is the Empire tech to up level?

I was of the understanding that you needed to go to enough huts of your level to be able to "level up" to be able to use the higher level huts.  I was doing this when I encountered the bug for my post: (v1.05) Adventuring Huts disappearing, bug or feature?

Yes, the in-game guidance on this point is severely lacking. I also had to browse through this forum to get any idea what was happening and how my adventure levels were determined. *sigh*