You can buy Middle-earth: Shadow of Mordor straight away.
More detailed answer
There really is no reason to play the games in a specific order, they have very little in common. storywise, they all are pretty much completely independent from each other.
What I know about the games in this list:
Fellowship, Two Towers and Return of the king (2002-2003) are third person action games, which follow the movies fairly closely.
Battle for Middle-earth 1 and 2 are great RTS games, whereas the first is close to the events of the books/movies, while the second looks at other parts of tolkiens mythology. Both feature a mode, where you can conquer the whole of middle earth (similar to the total war series). Despite being a series, the games really don't have anything to do with each other in terms of the story they tell.
However as EAs license for LotR games expired it may be very hard to get a hold of those games.
War of the ring(2003) is not based on the movies, but the general tolkien license, it's a new story, but I haven't played the game.
War in the north(2011) is an action oriented-rpg featuring a storyline not covered in the main books or books. It's nice for a fan, but the game is nothing spectacular.
Guardians of Middle-Earth(2012) is a moba(think league of legends or dota), I haven't played it, so I can't tell anything more about it.
Shadow of Mordor(2014) is a 3rd person open world action game in the vein of assassins creed. The story isn't covered in the books or movies. It's a very good game with a great feature in the nemesis system, but like AC-games it suffers from being incredibly easy.
The big one is Lord of the rings online. It features a massive game world, a long sotryline quest series, which touch both new ground and elements from the main book series. It's a wealth of content to explore, but of course it is an MMORPG with all its drawbacks and trappings.
I can't say anything about other titles in the list.