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!