What is the best way to improve my collection without spending money?

Like @ThomasP said, but I want to add the following:

If you're going to play ranked in the long run, craft early on one good cheap deck (usually these are aggro decks, like Pirate Warrior) and try to reach rank 5 every month. That alone gives you 505 dust (1 golden epic, 2 golden commons, 5 dust) for more crafting. Preferably, this deck should have several cards in common with other archetypes. I say one cheap deck because crafting cards is usually a waste of dust if you later open the same card from a booster, so you should only craft cards that you're going to use right away.

You should balance Arena runs with buying Adventures (700 gold each wing). Adventures give generally good cards that you can't get anywhere else. With the coming Standard change you won't be able to buy Naxxramas with gold unless you have the first wing, so I would buy that one at least. If you don't want Naxxramas cards you can disenchant them when Standard arrives.

Also, when Standard arrives there are going to be some balance changes which could affect Classic cards, so I wouldn't disenchant many cards until they announce the changes. Why? Because when they make an errata you can disenchant for the full crafting cost (50 dust for a common instead of 5).

And always get the free booster from weekly Tavern Brawl.

Edit: replaced old Face Hunter list with a post Gadgetzan Pirate Warrior

Personally I think the best way is to complete the daily quests as often as you can, replacing the lower gold ones (clicking on the "x" to get a new one) for hopefully more valuable ones. Every 100 gold, buy a pack. Also make sure that you win at least one Tavern Brawl each time it's available, as you get a free pack for that as well.

I used to save up the gold for for the dungeons, but I did some research and found out that the cards I would receive in most cases weren't worth the cost. (Except for the Grim Patrons so far), but I've not yet researched all of them.

Every time you open a pack, you of course have the chance of getting a card you already have 2 of, so the 3rd one is useless. This should be turned into dust. After this happens often enough, you should be able to craft some pretty decent cards.

Regarding legendaries vs rares, it depends on what kinds of cards you need. If the legendary is going to be super useful, craft that, otherwise craft something you think will round out your deck. Some cards have much more utility than others, but it also depends on what kind of deck you're building.

I also tried the arena, but found out that because I didn't have a stellar deck, I would lose a lot and it wasn't worth the gold investment. Personally I wouldn't try the arena again until I could get at least to rank 10 or higher, symbolizing that not only did I have a pretty good deck, but I knew how to play it effectively.

Apparently my experience with the arena (as well as my memory of it) wasn't that great, so if you're pretty good at playing a particular class and win consistently, it might be worth a go. For me, the other option of completing quests and using the gold to buy more packs was more certain though.

TL;DR: I'd buy packs, unless you have a phenomenal deck already, then I'd try arenas.

You get the resources to get more cards by winning games: daily rewards, the every-three-wins reward, the pack-from-first-win weekly tavern brawl reward, and end-of-season ranked rewards if you reach at least level 20 in the season (with bigger rewards for 15, 10, 5, and legendary ranks).

Go to some hearthstone sites and read about "net" decks: decks that other players have created, tested, and written guides for. Find a relatively cheap deck that you think you will enjoy playing (class + playstyle you think you'll like), and work towards getting enough dust to craft cards specifically for that deck.

You generally shouldn't craft just specific cards. Instead, craft cards that are part of a whole-deck strategy. This holistic kind of deck building (which, for purposes of learning, is best to use net-decks) will get you the most wins, and as described above more wins get you the resources to build a wider variety of decks.

And then just play in whatever mode you enjoy the most. I'd try to get to at least 20 in ranked each month, but above that I would only keep at ranked if you're enjoying it. Check out arena, casual (you may have to play quite a few games before the matching system starts matching you with appropriately skilled opponents to target a 50/50 win/lose rate), and the tavern brawl in addition to ranked.