Just Starting: An Overview
You start off in the world at your default spawn point, which is situated at the center of the surface world.
A common problem new players have is using the Hotbar. This is the bar that is always at the top of the screen and has 10 blocks numbered 1 through 0. You can select these boxes either by pressing the number on your keyboard that corresponds to the box, using a mousewheel, or by manually left-clicking on it. Pressing Escape brings you to the inventory which fills up as you pick up items in the world. Many items in the world stack (stacks are either in sets 99 or 250). In order to place an object that can be placed in the world (Such as the Workbench or a Torch), you must put it in one of the hotbar slots and select it while the game is playing (you cannot do it while browsing your inventory). Once selected, you can left-click on an appropriate spot within range of your character to place it.
 Just Starting: Playing the Game
In the beginning you will want to avoid traveling too far from your spawn point, and would especially want to avoid getting close to Corruption. Instead you should focus on making a safe haven out of wood and a Workbench to place in it. You should also try getting enough material to make a sword (A Wooden Broadsword is easiest, as wood is plentiful on the surface and you only need 7). If you're fortunate, you might spot some Copper Ore on or near the surface of the world and if you do, make the Broadsword out of that. It is a matter of personal preference, but by and large you should avoid making Shortswords as they are only short range piercing weapons and provide no rear or overhead cover from enemy attacks.
Once you're comfortable with the combat in the world (Practice your techniques out on Slimes), you should start digging belowground. You should do this near your base so if you die you won't have too far to go to get back. You should also use some of the spare wood you have to make platforms in order to get back up, otherwise you would be forced to dig your way back up again to return to your base. Don't forget to make Torches from wood and Gel you got from killing Slimes. Without torches you won't be able to see anything underground unless it glows on its own.
On the surface and underground (especially in larger caverns) you will find Pots. These can hold various consumable resources like bombs, wooden arrows as well as Coins. You may run into a Chest. Chests at the top level of the underground are regular wooden chests and contain low level loot, but early on in the game they are valuable resources. You can also pick up the chest by 'breaking' it with the hammer once you empty it out. Chests can be placed on any flat surface and used to store your extra stuff. You should place a chest at your House to make it easier to get to.
Later on when you have enough Copper Ore or (if possible) Iron Ore or better, you can make a Furnace to make Ingots which you can then turn into Armor at a Workbench to attach to your character for extra defense (and Set Bonuses), and higher tier armors tend to have secondary effects in addition to defense. Be sure to put the armor into the Armor Slot in the menu, instead of the Vanity Items slot.
As you fulfill various conditions, you will eventually gain NPCs that move into Houses (or large enough and well equipped rooms within the houses) and provide various functions. The Nurse will heal your wounds for a price, while the Merchant will sell you some stuff and allow you to sell things to him. The first NPC to move into the first house you make is the Guide. Prior to version 1.0.6 the Nurse was particularly valuable since you did not automatically regenerate health over time without the use of a Band of Regeneration. A nurse is still useful for saving time by making it so you don't have to wait for your health to regenerate. The nurse can also heal negative status effects (Not tested at this point, but is logical).
For beginner potions, you can make a Lesser Healing Potion by combining 1 Bottle with 1 Mushroom and 2 Gel. It will restore 50 health. As of 1.0.6 you will have to wait a full minute before you can drink another healing potion.
Also, by collecting 10 Fallen Stars you can combine them wherever you are to make a Mana Crystal which will provide you with 20 Mana once you eat it. [[Fallen Stars are found only at night when they crash to the surface at random locations. If you run into a Crystal Heart underground, smash it with a Hammer and you can then eat the resulting Life Crystal to gain 20 more health points. After 400 total health and 200 total Mana the respective crystals will provide no further boosts. Unlike Life Crystals though, Fallen Stars have further uses as Ammo or Crafting Material.
Thus ends the beginner guide. There is a lot more to this game than this guide could ever cover, so go forth into the world and find out for yourself what is possibleand have fun!