What can I do to reduce the symptoms of eczema?

Moisturize; HC cream. For mild to moderate eczema, Hydrocortisone 1% cream (use a thin coat on the rash, twice a day, for 5-10 days) is cheap, available at most stores, and works well. A daily moisturizing cream used 2-4 times a day helps to heal eczema & to prevent return of the rash. Examples: cetaphil, cerave, eucerin. Some people avoid creams with lanolin, aloe, or fragrances due to sensitivity to the ingredients.
Moisture. Water and solvents wash away your natural oils. Lachydrin (lactic acid) is very expensive but effective non-prescription lotion.
Protect. Avoid things that make you break out, soaps & wetness. Wash your hands only when necessary. Wear gloves when needed. Wear clothes made of cotton. Bathe only with a small amount of mild unscented soap, such as dove. Keep the water temperature cool or warm, not hot. Use the medicine your doctor gave you. Use a plain moisturizer daily. Avoid scratching or rubbing the itchy area. Manage stress.