I found that a good portion of my makeup has to go or be repurposed to fit my red hair. Your taste/style may be vastly different so I will offer what works for me, feel free to disregard.
Id say stay away from reds, especially bright reds. Personally I’d stay away from bright colors (like neon pinks) I’d recommend dark plums, deep deep reds, brown reds, and nudes. Depending on your skin color that is. You could also do purples.
For bolder lips, I will go with dark red, blood red, dark purples, mid to dark browns and black (my favorite).
Reds are tricky in that you need a complimentary tone to your hair which might take some time to find. You will also need differed reds to match the fade of your hair. I have 3, one for freshly dyed, one for mid-dye cycle and one for the ‘why is it so faded I should have re-dyed two weeks ago’ stage.
For a less bold look, I lean towards light Browns, nudes and taupes (love OCC lip tar and pencil in Sebastian for this). Although I rarely favor “less bold” – people already stop me to tell me about my own hair (positive and negative comments), might as well go the whole way!
I repurposed my not-quite-right red lip products as brow products, as I usually do red brows. My natural brows are very dark and don’t need filling in so the color mismatch isn’t an issue.
Tip: when you want to see what colors other women are wearing, Google “woman with red hair” in Google images and see how their makeup looks.