Upgrade order matters because each tool costs ore and days at the forge. The watering can should be upgraded right before a rain forecast so you lose no watering days; the pickaxe unlocks deeper mine levels and geode breaking; the axe clears stumps and hardwood; the hoe and trash can are lower priority.
This guide lists exact ore requirements per tier, Clint timing so upgrades finish when you need them, and which upgrades actually change your daily routine versus which are nice to have.