Regardless of skin type, as a basic beauty law, everyone should moisturize the skin. Even if your skin is oily, you can still benefit from skin moisturizers. The only exception is acne skin, because skin moisturizers may also block moisture entering skin.
The skin condition will determine if skin moisturizer is needed. When the skin is tight or dry, it means moisturizing is required. But be careful not to over-moisturize, because this can clog pores.
The skin around the eye contains no fatty tissue, and is therefore very thin and susceptible to wrinkles. Eye skin creams or lotion can be used to protect eye skin.