How to be self-sufficient in the sky?

The renewable resource page on the Minecraft Wiki is a great help.

I did something like this on an adventure map ("Land Claster" or something like that) where you're thrown onto a floating island and there are a half dozen others you can build towards. I might recommend a pre-made adventure map like this so you can learn by experience.

As far as what you need to build stuff versus living in a cave; the easiest renewable building material is wood. You need dirt to grow it, but you never consume it. Metals, strictly speaking, are renewable, but you can only get gold from Zombie Pigmen (in the nether), and iron from killing (personally, mob grinders don't work unless they just weaken them) Zombies. In order to get into the nether to get gold you'll need lava, water and diamond (or loose blocks of obsidian). Lava and water can also be combined in a cobblestone generator to create infinite amounts of stone (using a slightly less infinite amount of wood).

Animals yes, and you'll want some grass to feed the sheep and be able to use bone-meal to force some tall grass to get wheat seeds or flowers. Bone meal is free; almost too easy in a sky map with limited spawn space; you'll be rolling in it once you get a grinder going, along with tons of absolutely useless gunpowder.

Sand/glass is another non-renewable resource, along with other minerals; gravel, redstone, lapis lazuli, etc. "Coal" proper isn't renewable, but you can make the next best thing; charcoal, endlessly with your tree farm.

• grass: of course you could build a dirt stair and wait for it to grow, but isn't there > a much easier way? something like bonemeal maybe?

Bonemeal will not turn dirt into grass. You can also use a tool equipped with the Silk Touch enchantment (Thanks to Raven Dreamer and Mr Smooth).

• animals: they are pretty hard to bring up there

All you need are two animals. Once you bring the animals, you can use wheat to breed animals.

• metals: I heard you can create stone putting together lava and water, but what about iron and carbon and other fundamental metals?

Using water and lava, you can create a cobblestone generator. You can also create an obsidian farm with the same materials.

Most resources that you'll require are renewable, like wood (tree farm) and food (wheat farm). However, some are not: metals (like diamond and iron), for example, are not renewable and can only be found by mining.

The bare minimum to be able to function in the sky is a cobblestone generator, a tree farm, and wheat farm. With those three facilities, you'll be able to have a virtually infinite food supply and stone tools.

Something else you may want to consider is a mob grinder. This will allow you to acquire more resources, such as arrows from Skeletons and gunpowder from Creepers.

To complete the aforementioned tasks, you'll need water. You should probably have an infinite water source in your sky platform.

tl;dr -

• You can breed animals.
• You can farm cobble, food, wood, and mobs.
• You should have an infinite water source in your platform.