Standard game mode
The adventures available for Standard are those released in the last two years.
Blackrock Mountain is the second adventure ever released in Hearthstone, with the first being Naxxramas (see below, in the "Wild game mode" section). 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 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.
One Night in Karazhan, released in August 2016, is the latest expansion at the moment of this edit. It offers the same card/wing ratio of LoE. Even if not impactful as the previous adventure, it still features many good cards.
Barnes is probably the most powerful card of the adventure. Adding to the already decent body the possibility to get a mini-Sylvanas or mini-Ragnaros so early can be match-defining.
Spirit Claws and Maelstrom Portal cemented the already powerful position of Shamans in the game, making Midrange Shaman the new king of the meta.
Wild game mode
At the moment of this writing the adventures not included in the Standard rotation (so only Naxxramas currently), cannot be bought in any way. Their reward cards anyway can still be crafted with the Arcane Dust.
Naxxramas was probably the best adventure to get as a beginner, since it gave you a lot of powerful tools for many classes. Unfortunately, with the arrival of the Standard format, its cards have been rotated out and are only available in Wild games. It still features some pretty great cards:
- Mad scientist is a staple in many Hunter and Mage decks. 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 before the Standard update had at least a card from this expansion, and most of them are still played in Wild these days.
So, for the actual answer:
Get the League of Explorers. There are many good cards for almost every class, some of which are very powerful and/or class defining (like Reno or Tunnel Trogg). While it's close to being rotated out of Standard, there are still some months before that happens.
If you're planning to play in Wild anyway and have some spare Dust, keep in mind that the Naxxramas cards are still craftable and among them are some of the most powerful cards in the whole game.