Hearthstone 101: How To Get Gold Without Winning in Play Mode

The primary way of earning gold in Hearthstone is through completing "quests": these are in-game tasks, most of which involve winning with a certain hero or heroes to complete, and almost all of which require you to play against people instead of AI.
You get a new quest every day, even if you don't log in (but with a cap of three), and you can reroll one quest each day (this requires logging in, of course).

New players might find it difficult to complete quests with their collection of only Basic cards when they have to compete with players with better card collections.
To (somewhat) remedy that, there are also quests that don't require you to win to complete.  Additionally, I've created a Basic-only deck for each of these quests to help new players complete them quickly.

Basic Hunter deck for Beat Down
Basic Mage deck for Destroy Them All
Basic Druid deck for Only the Mighty
Basic Rogue deck for Spell Master
Basic Warlock deck for The Meek Shall Inherit

If you're a new player with just a Basic collection struggling to win in Play Mode, consider rolling for the quests above and try those decks too!

P.S. Each week there's also a Tavern Brawl (in PDT it's from roughly 10am on Wednesday to midnight on Sunday) in Hearthstone: basically a mode where things get EXTRA CRAZY.  (AKA much RNG.)
Your first win in Tavern Brawl each week nets you a Classic Pack!
Sometimes for these Tavern Brawls you play with a premade deck instead of your own; since you can complete quests in Brawls as well, the Brawls with premade decks in particular are helpful to players with minimal card collections, allowing you to compete more easily with other players and complete those win-requiring quests.
(Caveat: You need a class at level 20 to participate in Tavern Brawls.)

P.P.S. There's actually another quest that doesn't require you to win IN PLAY MODE.  The quest is Total Dominance, and to complete it you need to win 7 games, but you can play in Practice Mode and the quest is worth 100 gold!  It's awesome.

P.P.P.S.  There's one more quest that doesn't require you to win in Play Mode...it requires one of your friends to win!  Also, you get a Classic pack (worth 100g!) instead of gold.  The quest is Watch and Learn!  All you need to do is spectate a friend, watch them play a game and win it.  (You don't even have to watch from the start, all you need is to see them win!)

Hearthstone 101: How to Adventure

Hearthstone has a game mode called "Adventures": they are fights against unique enemy heroes with special hero powers (played by AI) that, when beaten for the first time, unlock cards otherwise unobtainable.  (These Adventures can be found in the Solo Adventures tab, under Practice Mode.)

Currently there are two Adventures, Naxxramas and Blackrock Mountain, each of which consist of five series or "wings" of bosses, the number of which ranges from two to four.  After beating all the bosses in a wing you also unlock one or two Class Challenges, which involve beating a boss with a premade deck to unlock a class card.  Unlocking a wing costs 700 gold, and wings must be unlocked in order, top to bottom.  (You do not have to beat bosses in prior wings, however, only unlock the wings themselves.)

The two most important points to take from here are these:
1) Cards obtained from Adventures are ONLY obtainable from Adventures; some of these cards are great!
2) Unlocking Adventures costs gold, which you could also be spending on packs or Arena.

For new players, I highly suggest saving your gold to unlock Adventures before spending them on packs.  This is because Adventures guarantee specific, otherwise unobtainable cards, whereas packs are random; additionally, many Adventure cards are consistently played in popular, competitive decks.

Now that I've got that out of the way, let's get to the meat of this post: HOW TO ADVENTURE.
Or omg there's so many wings and bosses and cards which one should I do first why is life so hard /cry

Or less sarcastically, which cards from Adventures should I unlock first?

First off, I believe the first two wings of Naxxramas to house some of the best Adventure cards, regardless of what class(es) you enjoy playing.  The first wing of Naxxramas (the Arachnid Quarter) unlocks Haunted Creeper and Nerubian Egg, which are used in a variety of decks (my budget Zoo Warlock, for example, plays both!). The second wing (the Plague Quarter) unlocks Sludge Belcher and Loatheb, also used in many decks, as well as some cards played in specific decks (Unstable Ghoul in some Warrior decks, Webspinner in some Hunter decks, and Duplicate in some Mage decks).

tl;dr unlock Naxxramas wings 1 and 2 first

After unlocking the cards in those wings, I think the ideal Adventure progression order depends on what class(es) you like playing; as such, I've separated the rest into sections for each class, with the wings listed being a roughly ideal progression order for that class, as well as the cards that have synergy with that class or a specific class deck.
I've also put asterisks next to cards that are played in almost all decks for that class, and double asterisks next to cards that are critical to a specific, commonly played deck for that class.
(I won't be going into detail with card specifics, as that's too much information for one article.)

Druid

Naxxramas wing 3: Spectral Knight
Naxxramas wing 4: Zombie Chow, Wailing Soul
Naxxramas wing 5: Echoing Ooze, Shade of Naxxramas*, Kel'Thuzad
Blackrock Mountain wing 1: Emperor Thaurissan

Hunter

Blackrock Mountain wing 1: Quick Shot*
Naxxramas wing 3: buy to get to wing 4
Naxxramas wing 4: Mad Scientist*

Mage

Naxxramas wing 3: buy to get to wing 4
Naxxramas wing 4: Mad Scientist*
Blackrock Mountain wing 1: Emperor Thaurissan
Blackrock Mountain wing 2: Blackwing Technician
Blackrock Mountain wing 3: Rend Blackhand
Blackrock Mountain wing 4: Flamewaker**, Chromaggus
Blackrock Mountain wing 5: Blackwing Corruptor, Nefarian

Paladin

Naxxramas wing 3: buy to get to wing 4
Naxxramas wing 4: Zombie Chow**
Naxxramas wing 5: Echoing Ooze, Shade of Naxxramas, Kel'Thuzad, Avenge**
Blackrock Mountain wing 1: buy to get to wing 2
Blackrock Mountain wing 2: Blackwing Technician
Blackrock Mountain wing 3: Dragon Consort, Rend Blackhand
Blackrock Mountain wing 4: Chromaggus
Blackrock Mountain wing 5: Blackwing Corruptor, Nefarian, Solemn Vigil

Priest

Naxxramas wing 3: Deathlord
Naxxramas wing 4: Zombie Chow**, Dark Cultist*
Blackrock Mountain wing 1: buy to get to wing 2
Blackrock Mountain wing 2: Blackwing Technician
Blackrock Mountain wing 3: Rend Blackhand, Twilight Whelp**
Blackrock Mountain wing 4: Chromaggus
Blackrock Mountain wing 5: Blackwing Corruptor**, Dragonkin Sorcerer, Nefarian
Naxxramas wing 5: Echoing Ooze, Shade of Naxxramas, Kel'Thuzad

Rogue

Blackrock Mountain wing 1: Gang Up
Naxxramas wing 3: Dancing Swords, Deathlord

Shaman

Naxxramas wing 3: Reincarnate
Naxxramas wing 4: Zombie Chow
Blackrock Mountain wing 1: Emperor Thaurissan
Blackrock Mountain wing 2: Blackwing Technician, Lava Shock
Blackrock Mountain wing 3: Rend Blackhand
Blackrock Mountain wing 4: Fireguard Destroyer, Chromaggus
Blackrock Mountain wing 5: Blackwing Corruptor, Nefarian

Warlock

Naxxramas wing 3: Voidcaller**
Blackrock Mountain wing 1: Emperor Thaurissan
Blackrock Mountain wing 2: Blackwing Technician, Imp Gang Boss**
Blackrock Mountain wing 3: Rend Blackhand
Blackrock Mountain wing 4: Demonwrath, Chromaggus
Blackrock Mountain wing 5: Blackwing Corruptor, Nefarian

Warrior

Naxxramas wing 3: buy to get to wing 4
Naxxramas wing 4: Death's Bite*
Blackrock Mountain wing 1: Grim Patron**, Emperor Thaurissan**
Blackrock Mountain wing 2: Blackwing Technician
Blackrock Mountain wing 3: Rend Blackhand
Blackrock Mountain wing 4: Revenge
Blackrock Mountain wing 5: Blackwing Corruptor, Nefarian
(NOTE: the Patron Warrior deck using Grim Patron and Emperor Thaurissan just got nerfed.)

tl;dr sorry bro

P.S. You can beat all Adventure bosses with just Basic cards (the ones you start with and ones you get leveling a hero to 10).  If you need help, check out a site like hearthhead.com, lots of people put up Basic decklists for beating specific Adventure bosses!

Hearthstone 101: Easy Money for New Players

So you've just finished the six-part tutorial for Hearthstone.  (It's still six battles, right?)  Or maybe you've gotten to playing some games in Practice Mode.  Or Play Mode.  Who knows, I don't know your life.  Point is, ya basically just started playing Hearthstone!

What you really want to know is: HOW DO I GET MORE CARDS?!
I guess if you know what "Shop" means and you've clicked on that button, you know that you can buy a pack of 5 cards for 100 gold.

So now you're asking: HOW DO I GET GOLD?!
After you complete the quests First Blood and The Duelist (you get these after completing the tutorial), you'll start getting "daily" quests (i.e. 1 quest a day, with a cap of 3 at a time).  These net you anywhere from 40 to 100 gold each, and you'll see them when you log in.
Additionally, you get 10 gold for every 3 games you win in Play Mode (also with a cap of 100 gold for 30 games won per day).

However!  There are some ways to get gold/packs that may not be obvious to new players.
These are one-time quests (meaning you only get the rewards once) called achievements.  I italicized that, so they must all be impressive and consequently difficult to complete, right?  Not true!  Some achievements are easy, even if you haven't opened a single pack.

Here is how so listen gud.
(Or, you know, here are the achievements, their rewards, and how to complete them.)

Ready to Go! (100 gold) - Unlock every Hero.
You can unlock a hero by beating it in any mode, though Practice is easiest.

Level Up (1 pack) - Get any class to level 10.
You gain experience and level up a hero simply by completing a game in any mode using that hero.  Mage is probably easiest since you level it up some in the tutorial, but eventually...

Got the Basics! (100 gold) - Collect every card in the Basic Set.
This actually means get every class to level 10, since you unlock new basic class cards every even level until 10.  Some of them are really good, so please do this!

Crushed Them All! (100 gold) - Defeat every Expert AI Hero.
You unlock the Expert AI in Practice Mode after either beating all the normal AI or unlocking all the heroes.  (I unlocked all the heroes by beating the normal AI, hence my uncertainty.)  This is the only achievement that you might have to work a little harder on, but the Expert AI is definitely beatable with just the basics!  (I also suggest keeping track of which Expert AIs you've defeated while you're completing this one, since Hearthstone sadly does not keep track for you.)

There you have it!  And have no fear of being matched with some guy who decided to swing his legendary-stuffed deck in Casual, for you need not step into Play Mode at all for these achievements.  :D

P.S.  Got an iPad, or a friend with an iPad willing to download Hearthstone?
iPack (1 pack) - Play a game on iPad.
This achievement can be completed in Practice Mode too!