While card packs and entrances to The Arena can be purchased with real money, they can also be purchased with the in-game currency - gold. Gold is earned by completing quests, and a few of the early quests will also earn a free Expert Card Pack.

After playing one match in Practice mode, the first quest should pop up, and it's a good one - play one match in Play mode (a casual or ranked match against a real player), and earn a free pack of cards. It's not necessary to win, just play.

From there, the next 2 quests are also relatively easy and rewarding - Play 3 games in Play mode to earn 100 gold (which is enough to buy another pack), and level any character up to 10, to earn a free pack. Once again, it doesn't matter if the games in Play mode are won or lost.

From there the quests get a little tougher and less rewarding, and they must be played in Play mode or The Arena, but they're still the key to earning gold. Most will have to do with winning games as a specific class of Hero, or defeating a certain number of minions, or dealing a certain amount of damage to heroes.