First adventure for a beginner

Curse of Naxrammas Many mage decks seen in ladder (e.g. freeze mage) feature and can rely heavily on Mad Scientist, to allow for early board presence. It also enables you to play secrets without losing tempo. Some variations of freeze mage also make use of Sludge Belcher and Duplicate. If you are leaning more towards the style of play that freeze mage offers, Curse of Naxxramas is probably the way to go.

Blackrock Mountain Blackrock mountain upgrades the mages toolkit with Flamewaker. Flamewaker is seen a lot in spell heavy mage decks and tempo decks. Emperor Thaurissan is another noteworthy card, as it is featured in freeze mage type decks also.

League of Explorers For mage, League of Explorers offers less than the previous two adventures. However, cards like summoning stone and flaming torch have seen some play in ladder, though not in decks that are as viable as those such as Freeze and Flamewaker mage.

Overall it comes down to your playstyle, both Curse of Naxxramas and Blackrock Mountain have cards that feature in most viable mage decks, so either is a solid choice.

As a new player to Hearthstone, the two goals you should have are to open Naxxramas and buy exclusively Goblins vs. Gnomes packs with your gold.

Naxxramas has the most defining and useful cards across multiple classes including Haunted Creeper, Zombie Chow, Loatheb, Sludge Belcher, Mad Scientist and a few of the class cards are staples in many decks such as Death's Bite for Warrior, Duplicate for Mage, and Dark Cultist for Priest.

Goblins vs. Gnomes features the most broken cards in the game arguably and is the reason that The Grand Tournament cards feel so underwhelming. These cards include Piloted Shredder (most used Neutral card in the game), Dr. Boom (most used Legendary in the game), along with the most dominant tribe Mech being a majority of the cards. Because you mention playing Mage, Tempo Mage gets a good portion of it's cards from this expansion in the form of Flame Cannon, Clockwork Gnome, Unstable Portal.

Following this pattern of buying cards, you should have good approximations of top tier decks faster than any other card buying strategy.

• Most-used statistics taken from personal scraping of Hearthpwn.com

Now, after the recent announcement from Blizzard, I would say beginners should avoid Naxxramas, since the cards will not be allowed in Standard mode. Also buying packs from Goblin vs. Gnomes is not recommended any more, because they will not be allowed, either. Everything else from the other answers is still true.