Each randomly generated world comes with at least one floating island, which contain a chest with unique loot and are interlaced with gold and silver ore. Finding a floating island is hard as the altitude depends on the size of the world you created, but building a "sky bridge" can help. On small worlds, building the bridge at about 200 blocks above sea level will put you within sight of an island. Keep an eye out for vines floating in the middle of the sky, as that's the bottom of your island.
Everyworld has an Underground Jungle and a Dungeon swarming with very difficult enemies and very valuable items. They are found just before the "ocean" at the edge of the world. If you walk in the same direction long enough you will come across one of them. The other one will be at the opposite end. So, if you walk west and find the Underground Jungle, to the east you'll find the Dungeon. Take strong armor, plenty of potions and maybe even a friend or two as they can be quite difficult.
Equip the following equipment: gold helmet, a frost chest plate, and frost legging. Also have the black and blue flame dye on all armor pieces. You will receive special vanity effects, which is a reference to Red's armor as depicted in the 1.2 Random Happenings trailer.
The Eater of Worlds is a difficult boss if you're not prepared. A good strategy for taking it on is to build a large enclosed arena a couple dozen blocks above wherever you plan to summon the boss from. This will keep smaller enemies out and the gap will give you enough of a warning to allow you to avoid most of the boss's attacks. Bring a lot of Shurikens if you don't have any flail weapons yet, as they will deal the most consistent ranged damage.
Every night time stars fall from the sky. Collecting ten of these increases your maximum mana. On normal terrain this can take a long time and can be a risky in pitch-darkness. By creating a Sky Bridge (by making a ladder around a hundred or so blocks in the air and laying down stone or dirt in a straight line), you can create a long stretch of land for the stars to land on. With a speed-increasing item such as Hermes Boots or Necro Armor equipped, you can travel the length of your bridge while collecting stars very easily. Zombies will still spawn on the bridge and you might get the occasional harpy or demon eye, but you won't have to deal with rough terrain, chasms, water or anything that spawns from the corruption.