Guild Wars 2

How do i access my quest log?
How do i get rid of old quests?

Would appreciate some feedback, thanks!

Hero Tab, then Story Journal, then choose the Story line you wish to have active.

Finish them? Choose a different Story line?

Good luck.

There are no questlogs in this game simply because there are not quests

What there are is heart quests which you make in the area were you are. When you have filled the quest bar in your top right corner you are done and automatically get a reward for it.

On your top right screen you also have your personal story showing. This you can disable in your options. This wont remove your story but it is not showing on the screen. You should make the personal story anyway, it is a huge part of leveling.