@Rarebear explained white balancing pretty well. However, I would suggest you not use the automatic white balancing settings since those can still be a little off depending on your settings. Usually, all you need is a white piece of paper and a button on the camera that you press and hold for around five seconds or so and then your shot will be white balanced but it’s important to follow the specific instructions for your camera. I only mentioned it because if you are doing makeup tutorials, color can be very important.
I don’t know about specific models since I haven’t used most cameras and the ones I have the most experience with are the really nice cameras we have at my school that cost thousands of dollars but the best thing you could do would be to go to an electronics store and scope out some cameras in your price range. If you find something you really like but that is too expensive for you, you can probably find it on amazon or in another store for cheaper.