Welcome to Terraria. If you are brand new to the game, and feeling a bit lost, here is a bit of a guide to get you through the first part of the game. To get the most out of Terraria, you need to take the game in steps.
Step 1) The first day.
On your first day, you have a number of goals you want to get done before the sun sets. These are:
1) Gather wood. About 200 wood is a good number. Use your copper axe to chop down trees for Wood and Acorns
2) Kill a slime. Slimes drop Gel, and you will need that for
3) Craft some Torches, and a Work Bench. You can craft items from your inventory page.
4) Build a House. To make a house, it needs to be large enough. At minimum, either 8x4 or 7x5, surrounded by walls, and with backwalls. Your first house is probably going to be made of wood. Just lay down wood to make walls, floor, and ceiling. After that, lay down your work bench and use that to craft a Wooden Hammer, a Wooden Sword, wood backwalls, a door, and a chair.
While you are digging around, be on the watch for Clay, Stone, and Ore. When night comes, go to your house and stay inside until dawn, unless you feel bold enough to try attacking the monsters of the night, Zombies and Demon Eyes. If you get a message saying Blood Moon however, get to your house and block the door. You probably can't handle a blood moon with your current equipment.
Step 2) Daytime surface, and Cave exploration.
Now, you have a house, and a base of operations, as well as a basic weapon. Now you need to focus on getting armor and a better weapon, and that means metal. Start exploring the surface during the day, and be on the look out for Copper ore and Iron ore. You will also need to gather some stone. If you find Purple grass (Corruption), or grass of a lighter shade of green (Jungle), turn back. Take note of where these areas are for later. While exploring the caves, look for breakable pots, and treasure chests. If you reach a point however where the dirt backwalls turn into a dirt and stone background, turn back for now. Your goal is to craft a full suit of Copper, or better, Iron armor. Also, get an Anvil(Iron bars), Furnace (Wood, Stone, and Torch), Bucket(Iron bars), Sawmill (Iron chain, Iron bars, Wood), Loom (Wood), and an Iron Broadsword (Or bow, if you prefer). If you manage to find a Boomerang and a Spear from Treasure chests, all the better.
Step 3) Night surface, and Underground exploration.
With armor, and a Iron weapon, you should be in good shape to fight off the Zombies and Eyes at night. Be on the watch for Fallen Stars. You will need at least 100 of those, but don't worry about collecting all 100 right now. Fallen stars are used to increase your Mana, which lets you cast spells. You probably don't have any spells yet however. You can also handle the underground without much trouble. You will encounter stronger kinds of Slimes, Skeletons, Bats, and Worms. If you find lava, your too deep. Always make sure you know the way back to the Surface. You are now looking for Silver and Gold ores, and you are looking for a Hook, which is dropped by Skeletons. Combining that hook with Iron chains gives you the Grappling hook, which will make navigating the caves easier. As before, keep a watch for Treasure chests, but you may also find Heart Containers. Break those with a hammer to increase your Max Life. Craft the Gold Pickaxe when you can, and a Gold sword won't hurt. Try to upgrade your armor also. With Silver armor and a Gold sword and Pickaxe, we go to the next step.
Step 4) The Corruption, The Eye of Cthulu, and the Eater of Worlds
Remember the Purple grass? Well, we are going there now. The mobs here are stronger, and all of them can fly. Get some good practice killing them, and collect the rotten chunks that they drop. Also look for Vile Mushrooms. Build a Skybridge over the Corruption, then take 6 Vile mushrooms and turn them into Vile powder at an Alchemy Station (Just put a Bottle on your Work Bench. You can use Sand at a Furnace to make Glass, then turn the Glass into Bottles). With the Vile Powder, and the Rotten Chunks, find a Demon Altar to craft Worm food. Then your all set to fight the first Bosses.