This one is super important, yet I rarely think to mention it...
Mark where you see surface bits of desirable ores, such as copper. There will be copper ore directly underneath the surface ore, usually needing to mine several blocks to get to it, but sometimes it will be the first minable block.
I mark it on the map, but I also mark the exact spot with a piece of grass (placed as if about to make a firepit).
If for some reason your map is stingy on surface copper for that first pickaxe and hammer, you can always pan sand or gravel blocks to get those last few nuggets. (knife + log = pan; panning requires that the player be standing in water)