Backpack: 4,200 Diamonds (Not including free bag unlocks); we are intending to increase available bagspace in the future, so this is subject to change.
Bank: 22,200 Diamonds (Not including free bank unlocks); also planning to increase available bank space in the future, so again, subject to change.
Lair: 14 Diamonds (And you get 2 free unlocks); we may be expanding this. You also get 20 slots at VIP10.
Stable: 10 Diamonds (And you get 2 free unlocks); we are also going to be expanding this so it will change eventually.
Dragon Chamber: Each level is 1 more diamond than the last, with the last set of slots being 31 Diamonds and 0 Diamonds.
The Backpack and Bank unlocks are in a pack of 5 for $2 in the marketplace (Utility -> Card Expansion), but you can buy bag and bank unlocks cheaply with Diamonds initially.