Terraria/Walkthrough

Upon creating a new character, you will receive three items: a copper shortsword, a copper pick-axe, and a copper axe. While this minimal equipment can get you through the day, your first priority is to allow survival.

Starting equipment
Even though you have a copper weapon, it is the weakest in the game with a limited range of attack, and doing little damage. To fix this, switch to the axe and chop down any large tree to collect the wood that falls down. Chop down a few trees until you have about 100 units of wood.

Switch to your inventory, and craft the workbench with the units of wood. Place the workbench, and create a wooden broadsword (which attacks in an arc, in addition to doing more damage), and a wooden hammer (which can remove background walls.)

Once you have this equipment, you will now need to build a light source. Collect some more wood from trees, and collect some gel from the slimes that jump around. With wood and gel, you can craft three torches. There is no need to worry about using too much wood - there is more then enough trees at the starting area, and you can regrow them using acorns (either dropped or purchased).

Building a base
Once you are able to build a workbench, you now need to build a small base. Make a box at least 40 blocks large using wood or any other materials you may want.

You may also need to clear the background as well. Switch to the hammer, and remove the existing dirt wall (if there is any), and replace it with a wall of your own. This wall is crafted at the workbench, and may be any material as long as you placed it yourself (wood wall, dirt wall, stone wall, etc.). Place light sources within the building, and outside.

You can craft a wooden door, a requirement for NPC housing (though they can be replaced with platforms). Doors are three blocks tall, so you should make a gap in the wall of your house to accommodate this. A Table and Chair will allow the NPC that spawns with you, The Guide, to move in.

Eventually, you will want to construct a bed to use as a spawn point. The Bed can be crafted with Wood and Silk, which is made by Cobwebs. This can be placed in any of the above rooms, and once used, sets the spawn point. It can also speed up time if waiting for day or night. If it doesn't set the spawn point, then there may be something missing (e.g. a small gap in the background wall, or there is a gap in the foreground).

Additional crafting
To create more advanced items and weaponry, you will need to construct the following:
 * A Furnace to melt ore into bars (requires 20 stone, 4 wood and 3 torches)
 * An Iron/Lead Anvil to construct metal items (with 5 iron/lead bars made from 15 iron/lead ore at a furnace)
 * A Sawmill to construct furniture (requires 10 wood, 2 iron bars, and 1 iron chain)
 * A Loom to create silk (requires 12 wood at a sawmill)

Once all five of these items are constructed, you should be able to survive for longer periods or perform any upgrades as necessary.