How should I take care of a small, deep cut on the bottom of my foot?

Depends. If it is fresh it might need stitiches. If not clean it well with soap and water, apply a topical antiseptic and dry sterile dressing. Watch for signs of infection, heat, drainage, pain, swelling.
How deep? Any cut that is deep should be examined by a physician. If you are unable to get into one then clean the cut with soap and water. Dry it off and apply an antibiotic ointment and cover it with a band-aid. If the cut develops a smell or looks angry and red then you should have a doctor look at it asap. If the cut is too deep sometimes stitches are required to allow proper healing.