As has been stated, the Tir Tairngir book from 2nd edition and the 4th edition Land of Promise are the key books you need for running in the Tir. System Failure (the book that ended 3rd edition) has some on the shakeup in the Tir, where the Princes got booted out. Some of the metaplot books from 4th also have updates on the actions of the Princes, but you don't really need to know that to start.
Critias is the first, best source for all things Tir, as he's the resident expert amongst the writers.
As far as the relationship between Tir Tairngir and Tir Nan Og, the relationship is strained, to put it mildly. The Tirs were both brought about by the Immortal Elves. The thing about the IEs, though, is that when it comes to working with eachother, they tend to be as bad as dragons in the 'my plots are better than your plots' thing, to the point where backstabbing and intrigue is like breathing for them. So while the Tirs would probably cooperate against the dragons, if it came to it, and would definitely move against Aztlan if they thought they could win, their own plots keep them as uneasy allies. But that's all on the uber-high level stuff. For most people, they see the difference as Tir Nan Og being a more mystical/cultish place, while Tir Tairngir tried to be a new take on medieval serfdom. One side thought the other were dreamers, and the other thought they had given up their paths on the wheel.
