I use these organic makeup wipes made for sensitive skin. I wouldn't advise using only a cleanser as a makeup remover. I've done that before and then when I dry my face on a towel I see all these foundation residue wiping off! So now I use makeup wipes and THEN my cleanser.
To take off Mac Studio Fix fluid, which is a high coverage foundation, I use Mac's Cleanse-Off Oil. My own jojoba/castor oil concoction also works well. While I use CeraVe for normal cleansing, the few times I wear makeup I find it is simply not strong enough.
For eye-makeup. Baby wipes. Just as good as "eye makeup remover" and tons cheaper and don't sting your eyes.
1. Baby wipes - If I'm really lazy. Perfume and alcohol free of course!
2. Simple Eye Make-up Remover - I'm not that fond of Simple, but this make up remover is really good. Because it is for the eyes, it's more gentle than a general purpose product.
3. Micellar water - This is good if your skin is sensitive, it's multi-purpose and can be used as a cleanser and a toner. The best thing is you don't have to wash it off! Just soak a cotton pad in it and wipe. Try searching for things like Bioderma online.