Now if you've been paying attention , you may have noticed that when you cut down some trees you got more than just wood. You also received an acorn or two (or ten). That's how we're going to make more trees. First equip the acorn, next find a good open spot in the sunlight for your tree to grow. Now just click on the spot you want to plant the tree and voila a sapling is born. You may need to clear the area of surrounding flowers or plants before a tree can be planted. This can be done with a sword, pick, or most other tools.
The trees need enough room on both sides of them in order to grow to their fullest potential. This means that two saplings directly adjacent to each other will not grow, neither will a sapling right next to the edge of a cliff. Make sure to space out your saplings so each one has enough room and they will all grow to supply you with wood ... and more seeds to replenish the garden.
If there is no good open spot in the sunlight for your trees to grow, make your own! You can use your pickaxe to level off a large section of land (preferably right next to your house) and a mallet to knock down the background dirt. Give a few hours for the grass to cover the newly plowed dirt and begin planting your acorns for your very own forest of replenishable wood.
The trees might be a bit smaller than the ones you cut down, they can grow. Anyway, in the meantime, let's go get some stone and metal.
After you got yourself a nice little(or a nasty large) home set up, it's time to go and mine some ores. Now, you can either choose to dig straight down (don't worry, it's safe enough, as you can see through walls with enough light) or explore a bit and find a nice suitable mine. Chances are that you have found some different colored stone whilst exploring, those are ores, mining them might be a good idea. Once you've collected about 15 iron ore, you can explore more or just return to your cozy house. If you've got a nice amount of stone, wood and torches you can make a furnace. Try placing it down just like you did with the workbench, but this time, it takes a 3x2 space. If you look in your crafting window after placing the furnace, you will see that you can make bars out of the ores you saw in the caves/overworld. Try to make 5 iron bars and take them to your crafting bench, with it, you can make a nice iron anvil. Place it down, and if you have enough wood and bars, you can make new tools/weapons! With this, you can upgrade the tools/weapons you have currently.(If you have enough materials) You can also make your own armor, but it will take a lot of bars to make a full set.
While mining underground you may come across some pools of water / underground lakes. Know that it is possible to use torches underwater, but it is impossible to place them. If you come across one of these pools, don't fear diving down underwater to continue moving downwards, you have a rather large air meter. However should you fear running out of air there is a technique to allow yourself to get more air underwater.
First either find a wall or make a small shaft at the bottom of the water. (The shaft should be around 10 squares deep) Then dig a 3 high 1 wide hole in the wall and then a 4-6 high and 2 or more wide hole in this wall. You should see that the water will not fill the upper squares. Now you merely need to surface, in that hole and your air supply will refresh.
You may also come across large deep chasms inside the caves. A good way to traverse them is to create a gapped bridge. To do this create a background wall of any material (stone and dirt will be in excess so those are recommended. Place a strip as far as you can reach then build the stepping stones at least 1 block apart. There are 2 benefits to this. 1 light can still reach below and 2, most enemies cannot actually pass through the blocks (you can't either so make sure to leave a somewhat larger gap near the end so you can jump down/up.
Notice in the last picture, the player has a time display under the item bar, more than 5 hearts and a star in the top right corner. The timer is given if you craft a Copper Watch at a workbench. The extra hearts are found via crystal hearts underground. You need a hammer to get them so its a good idea to bring one with you. Grab as many as you see (you can't go wrong with extra health). Lastly the star is mana power. This won't be used until later, but for now its good to start stockpiling. You craft mana crystals from 10 fallen stars and use them to receive mana power.
Try to keep this going for a while until you can make a nice set of armor (Gold or better recommended) and then we can talk about fighting your first boss!