Exploration and Resource gathering
Exploration and Resource gathering is probably the main focus of the game. So getting good at it is always a good thing since you will be doing this a whole lot. We've already covered wood. The other resources that are much more valuable are usually found underground in the form of ores. In order of value and rarity, Copper/Tin, Iron/Lead, Silver/Tungsten, Gold/Platinum, Demonite/Crimtane, Meteorite, Obsidian, Hellstone.
Each ore gets tougher to mine with weaker pickaxes the more value it has. Some ores and objects can only be mined by strong pickaxes.
(Note: Ores like Copper/Tin means a world can have only one of that ore. Copper OR Tin. They are both functionally similar and the items they craft are also similar to their counterpart)
Next, how to dig. There really isn't a surefire way to mine that will always yield a bountiful haul. But you can increase your likelihood of spotting ores by making your tunnels more efficient. When making a mining shaft, it is advisable to dig straight down and branching off to the left or right when you spot ore or just feel that there is ore around. Exploration is key.
Quick tip: You can dig around at your maximum reach and plop a torch down to "see" a bit further.
--
Quick Tip: If you find a life crystal statue (seen in the above picture, the heart statue underwater). (which when mined gives a life crystal), mine it and keep the life crystal in storage. Don't use it yet because having more HP is a trigger to some of the more difficult events.
An alternative to digging is just wandering off to find caves that have pots that may or may not have items in them. Chests that have a variety of loot. Just remember the common rule of thumb is the deeper you go, the better the items.
Quick Tip: During your exploration, if you see a goldfish, kill it to obtain it. Spawning in a world where you spawn near water is an advantage. Since you can farm goldfish for food later on.
Quick note on Corruption / Crimson
Each world created can have one or the other. They are generally the same and have different variants of enemies. No difference in difficulty. You'll know which one you have in your world by running to each end of the map.
Corruption is a purple biome and Crimson is, as the name suggests, on the red side. Corruption has Demonite Ore as its special ore. Shadow Orbs as the breakable object, & the Eater of Worlds as its boss.
Crimson has Crimtane Ore as its special ore, Crimson Hearts as its breakable objects, & Brain of Cthulhu as its boss. The Corruption has a more of a Sharp-like / Hashtag pattern to its caves ( ###) and the Crimson usually takes the form of a large asterisk * with a huge center and branching caverns around it.
Another essential part that you should remember to do sometimes is occasionally run across the map above ground at night. The objective of this is to gather Fallen Stars. Fallen Stars are essential if you have any plans of making a magic wielding character. Craft 5 Fallen Stars into a Star Crystal that boosts your MP by 20.