If I were a new player I would do one of two things.
The first option is to learn arena. This won't happen instantly. It requires a real solid grasp on the basics of the game, like trading efficiently, knowing when to try and get value (like taking out 2 of their cards with 1 of yours) from a card and how, and knowing how to play around and against certain cards.
You'll also probably want to follow a guide on how to draft for arena. Heck, just read a couple guides on how to play arena. Once you learn arena though, you have the best way to earn cards for constructed at your fingertips. Just play 1 or 2 arenas per day. You don't even have to finish that day. Save up all your gold but open all your packs. Don't dust anything because you will get cards you want to craft and in the long run you make the most out of your dust.
The second option, which is more for card game veterans or people who have a very solid grasp on the basics, is to work towards building 1 or 2 decks and playing them on ladder. This isn't the best way to earn cards and definitely not the most efficiently. You're better off earning cards in arena, but if you like constructed or hate arena, you can build a deck or two that will work well in constructed and it shouldn't take a ton of time. But you will have to dust cards you might want in the long run.
There are two main classes being played that are low dust cost. You shouldn't feel bad or be made to feel bad for playing either one. Both have their skill sets, don't let people convince you it is skill-less.
Midrange hunter, face hunter, zoolock, aggro lock are 4 main decks that are pretty cheap yet very effective. The two decks of those 4 that have the broadest range of strong match ups are the midrange hunter and zoolock, although frankly I see these 2 decks fading from the meta over the next couple weeks imo. They'll not disappear though.
There are more than a few variations of those 2 deck archetypes.
I only really know the hunter decks, I don't really play zoo although I have tried them out in the past:
Here is a more aggressive midrange hunter
http://tempostorm.com/articles/say-hello-to-alesh-and-his-undertaking-sunshine-hunter-guide
Here is a trap based midrange hunter
http://tempostorm.com/decks/mad-hunter-with-gaara
Here is a beast focused midrange hunter (eh, for some reason I couldn't find it)
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Just a few thoughts:
I think we all started at your position (the new guy with no "advanced" cards) playing against seemingly unbeatable decks. Over time you'll find that perhaps you weren't losing just because someone was playing strong cards. You'll start to see all the plays you can make each turn and you won't really know which is the best play to make. This is where knowledge about your deck and your opponent's deck comes into play. And there are a lot of tells on figuring out what kind of deck you are playing against.