Yes, You can dig randomly and go straight down and you can find ores, the better ones are deeper, Some ores require more power tools to extract. The hammer is the best, If you hover over your hammer you will get a popup that tells you the percentage damage to each type of ore, over 100% is required to extract.

If you do go straight down, if there are any voids nearby that creatures live in they will be attracted to the noise and start towards you,