Naxxramas was probably the best adventure to get as a beginner, since it gives you a lot of powerful tools for many classes. Unfortunately, with the oncoming Standard format, its cards will rotate out and only be available in Wild games. It still features some pretty great cards:
- Mad scientist is a staple in every Hunter deck and heavily featured in many of the Mage's. For these classes it's a must-have.
- Haunted Creeper is played in many zoo/face/token decks (Druid, Warlock, Hunter). For the Hunter it has double value since it's also a Beast, enabling many deck synergies.
- Nerubian Egg is also widely used for its stickyness. Warlocks especially love this card.
- Sludge Belcher is one of the best Taunt cards in the game. One of the few cards in the game that I've seen being used by every class.
- Zombie Chow is very useful in control decks to stop early turns' aggro and has also some use in midrange.
- Loatheb adds to a big body the ability to disrupt your opponent's next turn
These are just a few examples, this adventure is filled with good cards. Almost every high-ranking deck on the ladder these days has at least a card from this expansion.
Blackrock Mountain is a little less "useful" than Naxxramas. The dragons you can get from this adventure are heavily featured in Dragon-themed decks (usually for Priests and Warriors). There are anyway some non-dragon cards that should be highlighted:
- Emperor Thaurissan a.k.a. Tarzan, a.k.a. "I won't survive more than a turn on the field" is considered by some the most broken card in the game, allowing insanely good combos to become reality. Lowering the cards cost is good, lowering it permanently is very good, doing it every turn while he's on the field is insanely good. Being the reward for the first wing of the adventure means also that he's very easy to get.
- Imp Gang Boss, while being a class card (Warlock), is probably one of the best cards of the expansion. Nice body, token generation, good interaction with the Warlock tools, it fits in many archetypes for the class.
- Flamewaker is the core of Mage's Tempo Decks and finds some use also in spell-heavy decks.
Most of the other cards in this expansion are situational and/or used only in very specific deck archetypes.
League of Explorers, the last one at the moment of this writing, offers the highest "card density" of any other adventure (45 cards for 4 wings vs 30-31 cards for 5 wings), giving you the best price/card ratio. This expansion singlehandedly created new deck archetypes and revitalized half-dead classes.
Reno Jackson has been one of the biggest surprises of this expansion. Considered by many a gimmicky card before release, has proved its worth many times after getting in the hands of the players. Many classes have now a Reno-themed deck, with Warlock being probably the one that puts it to its best use.
Entomb confirms once more the Priest as the most annoying class to play against. An unconditional removal that avoids deathrattles and gives its target to you is the source of a lot of frustration for anyone that faces a Priest.
Sir Finley Mrrgglton and Tunnel Trogg have rescued Shamans from the dark pit they had fallen in, letting Shaman decks (or at least one of them, Aggro Shaman) to be once again competitive on the ladder.
Previously my answer would have been to get Naxxramas as your first adventure, but after Blizzard's announcement of the introduction of Formats into Hearthstone, the suggestion to prioritize Naxxramas stands true only if you plan to play in the Wild mode. Otherwise, my suggestion is to get the League of Explorers, whose cards will be usable in the Standard format for at least another year.
Just keep in mind that, as soon as the new Spring 2016 expansion gets released, the Naxxramas adventure won't be buyable anymore, unless you got at least the first wing, in which case you'd be able to keep buying the remaining wings with gold. Naxxramas' cards will still be obtainable through crafting anyway, but they will only be usable in Wild mode.