This is what I have so far. This deck is based off of dealing as much damage as possible as quickly as possible. I have always felt that the best way to do this is with a sustained small amount of damage starting from turn one. The whole point of this deck is quick small hits. The Goblin Chieftain's ensure that should it be dragged to late game (which it should never be, unless I failed at my early game...kind of hard with nearly everything being under 3 mana), my little cheapies are going to be big sluggers. The direct damage ensures that even if my cheapies die/can't get through opposing defenses, I'll be able to bring the hurt. By turn two, with just two mana and the right cards, I could already have an opponent at 11. What I'm looking for, specifically, is this.

Can you think of any way to expand on its strengths and bring the cost down? Or is it best to stick with this? I don't want to make it weaker for the sake of cheaper. Just better, and cheaper if possible.

Lands:
20x Mountain

Creatures:
4x Raging Goblin
4x Goblin Guide
4x Warren Instigator
4x Goblin Chieftain
4x Goblin Bushwhacker
2x Goblin Shortcutter

Spells:
3x Quest for Pure Flame
3x Pyromancer Ascension
3x Punishing Fire
3x Burst Lightning
3x Lightning Bolt
2x Lava Axe
2x Fireball

Total # of Cards: 61

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All-out assault red is a time-honored strategy that has created its own paradox. As one of the oldest archetypes in the game, one would expect it to eventually boil down to one perfect list. But the nature of the Standard format constantly changing cards' legality, and the continual release of new cards, means that there is always some doubt over what's best to run. You're definitely close to an optimum build, but in Magic, there's always room for a little tweak.

Fireball, despite its iconic reputation, is not what you want. Fireball is best in decks that can put out a lot of mana, or that want to stall until they an unload as big a bomb as possible. Lava Axe is in a similar position-- it's a late-game card, a finisher. I say take them out and go up to four copies each of Lightning Bolt, Burst Lightning, and Punishing Fire.

I'm also not sure you want to run Pyromancer Ascension. To get the Ascension online, you have to be casting multiple copies of spells. Since you've got just as many Goblins as instants and sorceries, you just won't Ascend often enough to devote deck slots to it. A better choice for you would be Unstable Footing. True, it doesn't do any damage unkicked, but it does "counter" things like Fog or Safe Passage, and it allows you to deal damage to things with protection from red. Once kicked, it's an unpreventable Lava Axe. Why am I recommending what basically amounts to a Lava Aze when I just told you to take out Lava Axe? Versatility. Unstable Footing has other uses besides just "five mana for five damage", and it can defeat the kinds of cards that will give you the biggest problems-- Mark of Asylum, Safe Passage, etc. There are still some Fog decks out there, after all.

Quest for Pure Flame is another one I'm not sold on. It's good, but it takes time to charge up and you can't predict how much damage it will do. I won't pull it yet, since it has a lot of potential when you pull it off, but I'd like to suggest you watch it carefully. Are you happy with its performance? How often do you notice yourself hoping to draw it? When you draw it, are you usually happy or disappointed? How much damage does it usually do when you decide to cash it in? Goodness knows, if you can make the most of it, it echoes faintly of the Spinerock Storm decks that could get 21 damage out a single Grapeshot after laying Pyromancer's Swath. But does your deck exploit it properly?

Finally, Goblin Shortcutter is a lackluster one-shot effect that quickly degenerates into a 2/1. I'd rather you use Ball Lightning, which is still one shot, but it's a shot out of a cannon. Or Goblin Razerunners, holding the distinction of not dying to an opponent's Lightning Bolt. They're good even if you don't sacrifice lands to them, but if you do sack a land, even one, you've got repeatable, unblockable damage in addition to your 4/5 or bigger.

And finally, your lands. Mountains are fine, but a copy or two of Teetering Peaks or Valakut, the Molten Pinnacle would turn even your lands into sources of damage. No more than two copies total, though. They come into play tapped, and that could slow you down if you don't draw enough real Mountains. In your deck, I think I'd treat Teetering Peaks as a spell instead of a land.

Good luck!