Around the eyes, specifically? Makeup Remover and cotton balls. But apparently people love other people in eyeliner that on-point, so maybe keep it a day or two.
But, as others have said...
1. Yeah, moisturize your face before application, regardless of makeup type being applied. Except if its grease makeup. Skip the moisturizer and the grease makeup all together (Kidding, kidding).
2. Face washes and/or shampoo in general will do most of the work for all but the most stubborn of makeup (Often stuff like alcohol-based makeups).
3. Interestingly enough -- those Stridex/Clearasil facepads you probably used as a teenager cut through practically everything. Probably because they're made to destroy anything on your face, no matter the cost. Do not get this in your eyes.
4. Attend to specific spots with makeup remover specific to the type of makeup involved -- anything other than alcohol can use regular makeup remover, just about any brand.
5. Really, moisturize afterwards. You're best off doing a full face-routine (Cleanser -> Toner -> Exfoliant -> Moisturizer -> Antioxidant) before and afterwards, but moisturize at the barest minimum.