For everyday makeup, I use two things: a light foundation/concealer (RMS beauty Un-Coverup) and a solid mascara from T.W.I.N.K. beauty on Etsy. Neither has any plastic packaging, and both use safe ingredients. RMS has other products, if you are looking for eye or lip color, or you could try your hand at homemade cosmetics.
(If you haven't seen solid mascara before, you wet the brush and rub it over the mascara to get the color on the brush).
If you aren't familiar with Plastic Free July, take a look at this post. Basically, it's a month-long challenge to reduce the amount of disposable plastic you use. It's a great starting point if you just want to give cutting out disposable plastic a try.
My first Plastic Free July was in 2013(!), and this month I'd like to highlight some of the things I do to reduce my plastic use, and some struggles. Many things are part of my routine at this point (buying in bulk, refusing plastic silverware, etc.), but I am not perfect, and I always find this challenge to be a good time to refocus and look at what trash I am actually producing.