Soak your foot in warm water to soften the tissue. Then pare away carefully with a sharp blade, or smooth away with a pumice stone, callus file, or even sandpaper. Corns and calluses are reactions to pressure and friction, so look for the cause. If you find tight spots in your shoes, stretch them or take them to a shoe repairer or a pedorthist to be stretched. You can also try wearing thicker socks or loosening shoelaces, or apply pads such as Moleskin, lamb's wool, or Second Skin to relieve pressure.
Warts look similar to corns but are caused by viral infections and can spread by contact.
Warts tend to grow inward and can contain folds of healthy tissue, so avoid cutting or abrading them. You can ignore small warts that do not cause pain, and they will probably disappear within a couple of years. A favorite site for a wart is the ball of your foot, and the pressure can cause pain. You can treat a plantar wart with salicylic acid preparations from the drugstore to kill infected tissue. Apply the preparation for a couple of days, then scrub the area with a small brush and soapy water for a minute, twice a day for two days.