If you have a dirt road near your house you can dispose of the fireplace or wood stove ashes in the road and it will just mix in with the soil.
Or you can also dig a small hole in your yard and dump the fireplace or wood stove ashes in the hole and bury them.
If that is not an option then just allow the fireplace or wood stove ashes to cool down completely and then place them in a garbage bag.
The fireplace ashes or wood stove ashes as long as they are cold and not smoldering can go into the trash can without a problem.
You might want to place the ashes in a bag and tie it off to prevent the ashes from going through the air and creating dust.