Until you have a pickaxe, you will have to scavenge the surface for copper ore in loose stone or pan for some ore nuggets. You can pan sand and gravel using a wooden pan, crafted using a piece of log with a knife beside it in the crafting grid. You must use only dry sand and gravel - muddy gravel won't work. The block needs to be on dry land (not touching water). You yourself must be standing in water. Right click the gravel or sand block to fill the pan, then hold your right mouse button to wash the gravel. You can pan 8 times from one block of gravel or sand. Once you collect a few dozen ore nuggets you are ready to leave the stone age.
Note: Surface ores are always an indicator of near surface deposits. If you dig just below where the surface pieces were, you are very likely to find a deposit.