The desert biome is one of the most difficult to survive and prosper in. If you spawn in a desert biome, you will most likely want to leave that area and relocate to a more hospitable and productive biome.
For basic survival steps, so that you can live long enough to attempt to explore your way out of the desert, refer to the Immediate Survival section of this Tutorial. That tutorial is about the voluntary challenge of playing in an endless desert, but the immediate survival tips apply equally to a player who has been unlucky enough to spawn in a desert, and just wants to survive and leave. However, if you are looking toward the challenge, then just keep on reading!
Where to Start
First of all, dig up a lot of sand to craft into sandstone, as that is the best material available for the moment. Focus on building a more secure house than you normally would, as it will take a long time to find enough wool for a bed. Prepare for the possibility of an emergency shelter on your first night. The two best are the Three-Down-One-Up and a jump pillar. Food may not be an issue if you can effectively hunt rabbits. Otherwise, look for any naturally generated structures that may contain a chest. They may contain useful materials. Deadbush is quite a common item in desert biomes. Use them to craft sticks. Then, if you find a desert village and it has a blacksmith, check the loot. You might find some iron. If you find 3 ingots, craft a pickaxe and mine some stone later on. Then, load up on cooked rabbit and then you can go mine and proceed with your game.
Setting Up Shop
Farming will not be available for a while. You may have possibly found dirt underground mining. Keep some of that. Carrots and potatoes can be rarely dropped by zombies. Desert temples generally contain iron ingots, or you may find some mining. Sticks are dropped from dead bushes. This eliminates the tool issue. Once you have a steady supply of food, you can now focus on more challenging tasks.
Fighting mobs in the desert is a little challenging, as the lack of cover leaves you exposed to skeletons and other ranged attackers. The lack of cover counter-intuitively makes it easier to find a target, and kill it. Creepers will be easy to kill in the open terrain. Bows are good weapons in the flat, unhindered terrain. Endermen, one of the more important mobs to kill, will be difficult, due to the fact that flat areas will leave them plenty of room to teleport. If you have the wonderful liberty of being near a desert pool, jump in and attack him from the middle. Husks can also be a problem due to the fact that they give you food poisoning.
Finishing the Game
Once you have everything you need, you can go and complete the game. Once you locate the portal, you can pretty much just beat it like any other game.