How to F2P. You only need 3 tips.
(1) Log in and complete your daily quest every day, but ONLY if it's 50+ gold quests.
If you draw a 40 gold quest, mulligan it. If it's still 40 gold, don't do it. Save it for tomorrow and draw a new quest and mulligan the old 40 gold quest. If you get another 40 gold tomorrow, mulligan one of them. Statistically, you have 2/3 chance of moving up to a 50, 60, 80, or 100 gold quest. If you REALLY want to min/max you should mulligan 50's, too, but that takes a lot of quest juggling and most people don't have the patience for it.
If you complete your daily quest every day (even if it's for 40 gold), you will pick up enough gold to randomly draw 10 Legendary cards per year (about once every 4-6 weeks). The quests are designed so you average 50 gold per quest if you just take whatever they give you. This is enough for 1 free pack every 2 days. The Legendary drop rate is tuned so that over the course of a year, you'll average about 10 Legendaries. That's not an accident, either. Blizzard designed it this way. F2P = 10 random Legendaries per year.
You can game the system a little by mulliganing 40g quests and only finishing 50 gold or more quests. Through careful min/maxing, you can bring your average gold per day on quests up to around 63-65. It winds up meaning about an extra 25-35 packs per year, which means around 1-2 extra random Legendaries.
Additionally, once you have most of the cards, you're going to get a lot of dust. Assuming you only ever get the minimum of 40 dust per pack, that means you accumulate enough dust for 1 legendary for every 40 packs. If you're F2P, that takes 80 days, which means you'll have enough dust for about another 4-5 Legendaries per year (365 days / 80 days = 4.56 Legendaries, but, you'll average more than 40 dust per pack).
Tavern Brawl is also good for 52 packs per year, which should yield about another 3 Legendaries during the year.
So there's basically 18-20 free Legendaries per year. That's enough to make at least 5 viable decks. It also doesn't take into account any extra gold or dust you earn from Blizzard giveaways or Arena.
(2) Classic is your foundation; don't chase expansions until your Classic card set is built up
You're going to suck early on. For one, you don't know how to play. For two, you don't know how the cards work/interact. For three, you don't know the deck archetypes. For four, you don't have enough cards to tech around them. For five, you don't have many cards to work with. And I could list a thousand other reasons. You are mostly not going to have the resources to chase the meta, so don't bother.
Focus on building your skills, learning the game, and mastering the classic set. This is because those cards are always going to be in the meta, and you need to know them inside and out. If you're just chasing the latest expansion, you're going to have a bunch of cards that don't work well because you're missing the classic cards you need to really get the most out of them, and that expansion is going to be replaced by another one within 3-5 months. By the time you get a deck put together, a new meta might obliterate it, and those expansion cards are just going to rotate out.
With some exceptions, most decks remain heavily focused on classic cards. For one, it's the biggest set, but moreover, the classic cards are, individually, generally more powerful than the expansion cards and they establish the basic playstyles of each class. Un'goro did chagne this a bit - I think the modern meta is more expansion-heavy that prior metas, but it's still around 50% classic cards in most decks.
So focus on getting classic cards at first. I spent my first sixth months buying almost exclusively classic cards, and just building individual expansion cards that I wanted. Stick with classic until you've at least got at least 1 copy of most of the rares.
3. Know Thyself
Do you have the patience to wait it out and slog through frustrating games? Yes? Then F2P is for you. With time and patience, you will build up a competitive card set and learn the game and be able to go toe-to-toe with the guy with 20 golden Legendaries and 9 golden heroes.
If not, then do yourself a favor and go pick up a few extra shifts so you can slap down the clams to buy some cards. If you're too impatient to play the long game, you'll just drive yourself nuts and feel like you're always behind, and you'll resent people who pay for packs, and you'll feel like it's completely unfair, and you won't enjoy yourself at all, even if you've got a 60% win rate. Because you won't remember the games you won because you put in teh time to learn the game and with your mediocre deck, you outplayed a guy who dropped $200 on an expansion but doesn't know what he's doing. That won't stick in your craw. What will is when you lose a game because a guy who dropped $200 has better than you and misplayed them all but still won.
So don't drive yourself nuts. If you like the game and aren't the patient type, just buy in.