I got into the Beta thanks to a friend who had spare keys :D
I beat the tutorial, can't wait to delve deeper in the game. Just want to do drafts, I'm assuming I can grind ingame currency for it like Hearthstone, right?
Yes you can get gold by playing the game, gems being the currency you get for real money.
You can enter drafts with gold or gems, but to earn gold you need to play matches, so it seems difficult to do drafts only while not paying real money.
My advice is to still have about 3 decks to go into free play, with all 5 colors represented, play a bunch of games when you connect, using the decks that will make you get the daily rewards.
Those are usually as simple as "Cast 20 Red or Black spells" and are quite rewarding.
In about 4 matches you'll easily have about 1000 Gold.
You have preconstructed decks to use, some are ok to begin with and easily upgradable, from what I remember the explore, Izzet and cats decks are among the best.