Sky bases give you the freedom of the entire sky, away from earth-based menaces and almost completely safe. However, they're quite dangerous to build and have a few notable problems. This page shows you how to build them without as much risk.
Each block receives a score on flammability, strength, and expensiveness. The lower the scores for flammability (F) and expensiveness (E), the better, while strength (S) is the opposite. Each is rated from 1 to 10, with 10 being highest. Note that a "1" in "Flammable" means that they can be lit on fire, although they cannot burn.
It's best to establish a method of ascension to your base first, as you might be going up and down to collect resources while constructing it. The four notable methods are:
Staircases are the least compact, but they also are an aesthetic opportunity and work well. Just remember to add a railing! They are also the most expensive.
Water is one of the easiest, but a special collection pool is required at the bottom if you don't want it spilling all over. It's also the slowest, as the downward current is hard to climb.
Ladders are quite easy to get with an abundance of wood. However, they require a platform of solid blocks to rest on. Also, you can't just jump down into your shaft, as ladders have a hitbox.
Vines are probably easiest. While they don't have a hitbox, only one is required - they can just grow down over any block and form a ladder. You can also just jump in - they don't have a hitbox that you can collide with. Note, however, that they grow only in Swamp Biomes or Jungle Biomes and require 2 Iron Ingots in the form of Shears to collect. Growing also takes time, so if you don't want to wait for your ladder, you'll have to get more of them.
You may want to put some useful resources in an Ender Chest on the ground, so that you can retrieve them at will from an Ender Chest in your soon-to-be base. This makes it so you don't have to run up and down getting resources from normal chests.
The Basic Base
You want to start by making a platform of whatever material that is suitable for you. Make it in whatever shape you want. Just make sure it's flat. You'll want to put all your necessary materials (Chests, Crafting Table, Furnaces, etc.) on this platform for easy access.
The walls will offer some protection and hold up the roof. You may want to make them at least partly transparent so that you can have windows, but just about any solid material will work. Note that you'll want to put a door in the walls. An aesthetic opportunity is to use vertical Wood blocks to do this, as the bark creates a nice "wallpaper."
The roof should be tall enough to allow you to walk in and out comfortably, but not enough to allow Endermen to teleport in: as such, it should be two blocks above the ground. It can be made of whatever material you want.