I've been to the Quest Log and I've been to the Garden of the Magnates (which is where I've just finished)... I already accepted the next quest but for some reason I'm unable to complete the quest and get the rewards. How do I complete it? For some reason, the letter never loaded and I had to refresh the page a couple of times just to do anything.

Also, I've got another quest, the one from Fungus Grove, that I could never complete but I'm all the way at the Forgotten Tomb. How can I complete that one?

You can complete the Fungus Grove quest by clicking that link at the bottom to switch to the "old" quest log.

As for the other quest, once you've met the conditions of the quest, the complete button should be clickable. Click it to complete the quest and get your rewards.

With the "new" quest log, once you complete a quest, it will stick around in your log as "Completed". This is so you can re-read what you've done so you can remember the story.