What is the best way to keep facial skin hydrated? By consuming plenty of water or using a moisturizer?

Both. And i will add a third: increase the humidity of the surroundings. Air conditioners and winter heat dries out the air. Ambient moisture is a nice adjunct to better more comfortable living. Drinking plenty of fluids, using moisturizers, creams and lotions to lock in water and prevent evaporation and using whole house or individual room humidifiers can help with moisturizing the skin.
Both. Keeping your body well hydrated will help keep facial skin moist as well. Moisturizing lotions/cream/unguents (in increasing order of moisturizing effect) prevent dryness. Washing the face gently with the hands (cloths/brushes/loofas abrade the skin) and mild, moisturizing soap (dove, for example) keeps the skin soft & moist. Avoid very hot showers as they strip skin of natural oils & dry the skin.