Is it this?
You are discussing the Chronicles of Narnia, in the chronological order of the books.
My first has green and yellow
To take you there and back.
The Magician's Nephew
Yellow rings were used to travel to the magical world of Narnia and green rings to return.
My second is made up of three,
A king, a queen, and just one more.
The Lion, The Witch And The Wardrobe
The lion (Aslan) is the
rightful king of Narnia king of the jungle, Jadis is the White Witch who pretends to be the queen and one more points to other object in the title, which is the wardrobe.
My third is all about a message,
Conveyed in the nick of time.
The Horse And His Boy
A lost prince is rediscovered and it's his timely warnings that prevent an invasion.
My fourth is north of rain around,
A king in the making there.
Prince Caspian is the king in the making.
EDIT: As per rand's comments,
"north of rain around" again seeks to sneak in the name of Narnia in anagram form. "rain around" is a clue to find an anagram for rain, which is the country Iran. And the Caspian Sea lies to the north of Iran.
My fifth is completely out at sea,
Travelling to the sunrise bright.
The Voyage Of The Dawn Treader
Prince Caspian sets out on a voyage by sea to find the seven lost lords of Narnia.
EDIT: The sunrise bright is a reference to the titular ship, Dawn Treader.
My sixth has giants and a seat,
And takes place in the north.
The Silver Chair
A journey is taken to the giant lands to the north of Narnia. The seat is the titular silver chair.
My seventh is a tale of war,
And tragic in the telling.
The Last Battle
A huge battle takes place that ends in the destruction (and re-birth) of Narnia.
The title is misspelt intentionally
because the letters '..an rain..' can be rearranged to form Narnia.
To conclude, I would like to apologize
since I haven't personally read the series, so please feel free to correct my explanations.