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 reclaim the workbench, or 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. Head somewhere slightly off of the starting point (by at least 1 screen), and clear out a room. Surround a desired area with blocks other than dirt - a cheap source would be wood, but you may also use the stone blocks.

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.

Crafting Wooden Doors wouldn't be bad too, because you can right-click it to open and close it. 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. If it doesn't set the spawn point, then there may be something missing (e.g. a small gap in the background wall, or the foreground blocks are dirt instead of brick).

Additional crafting
From what you have, you can only craft wood-based items. 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 Anvil to construct metal items (with 5 iron bars from 15 iron ore)
 * A Sawmill to construct furniture (requires 10 wood, 2 iron bars, and 1 iron chain)
 * A Loom to process silk (requires 12 wood)
 * A Alchemy Station to create potions (requires a table, chair, and a bottle)

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