Pickaxe progression is really a sequence of mining-power breakpoints. A new tool matters most when it lets you mine a resource that the previous tool could not reach.
Early-game upgrades
Replace the starting tool when mining speed or resource access begins to slow normal exploration and crafting.
Pre-Hardmode breakpoint
Before the Wall of Flesh, make sure your pickaxe supports the ores and materials needed for the equipment you still want to craft.
Early Hardmode priority
Hardmode introduces new mining requirements and resource tiers. Tool progression becomes part of survival because stronger gear depends on access to new materials.
Do not upgrade only for speed
A small speed improvement is useful, but a mining-power increase that unlocks a new ore tier usually matters more.
Pair tools with mining support
Mining potions, mobility and inventory space can make a run more productive even when the pickaxe itself is unchanged.
Stop when the next upgrade is not useful yet
If your current tool already mines the resources you need, spend time on armor, weapons or bosses instead of upgrading for its own sake.
Source: Official Terraria Wiki — Pickaxes.