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First let me say "I LUV Heroes of the Storm! ". Well designed, lots of character depth as well as player progression. While I am not a F2P player. Bought some Heroes as well as stim packs. For the F2P player getting that hero you want may be daunting.
So here is my list of tips for you new players that have a issue with money. Well money you actually have to spend.
1. Decide what hero you really want to work towards.
Each player has a play style that suits them. Your initial free characters may not be the one that suits you. So find one that is not necessarily on the free list(Or can be) and test it til you find one that you like. There are heroes from Starcraft, Warcraft, and Diablo. As well as the retro "Lost Viking" character!
Each hero is deemed for a specific purpose. You have ranged, melee, assassin, specialist, healer, support... Find what fits your fancy!.
2. Play each free character to level 5.
Each time you level a hero to level 5 you get 500 gold. You initially get 4 free characters to try out. So level them to 5 to get a whopping 2k of gold. Heroes also cycle! So the next day you may not have the same free to play heroes.
3. Player progression. As you level up your heroes you also have a player progression table you level up as well. There are key marks with player progression that offer you even more gold. If you progress to player level 10 and used the following tip above. You should be able to buy any hero that is 10000 gold or less.
4. Daily Quests!
Just like Hearthstone you get a daily quest each and everyday up to 3. Each quest offers gold as a reward. Make sure to do them often! As you can hold only 3 quests maximum at a time!
5. New Player stim pack.
As a new player you will get a 3 day stim pack. Make sure to activate it. The gold bonus for finishing a match is nice. 75 gold if you win. 50 gold if you lose.
Now some nifty things..
At player level 12 you receive a World of Warcraft pet as well as a very cool looking Hearthstone cardback. Some of the mounts and new skins you get are very cool looking as well!
Enjoy the Game! It is very fun! Hope the tips help!