How can I avoid allergic reactions to skincare products?

Skin allergies or dermatitis in response to using a skincare product is related to the ingredients.  Always read the list of ingredients to understand what you are putting on your skin.  It is also helpful to know the scientific names of certain ingredients, i.e. Vitamin E is identified in the ingredient list as Tocopherol.
You may have sensitive skin, and without pinpointing a certain ingredient, you might be senstiive or reactive to a product.  Bottom line: everyone is different and everyone's skin reacts differently to different products.
Do a "patch-test" of the product on a small area of your skin before using the entire skincare product.  For example, before applying your facial moisturizer, dab a small amount on your cheek to test it out for an unfavorable reaction, before you use the product on your entire face.  If we are talking about a body lotion, then apply a small "test amount" to your forearm and wait a couple hours to see if you agree with the skincare product.
Natural ingredients are the gold standard!!  In fact, you can make your own ingredible skincare recipes and trust that it is chemical-free. Natural skincare is the way to go!

The majority of allergic reactions to beauty products are due to preservatives, with fragrance being a close second. Almost any beauty product you use has the potential to irritate the skin on your face. This means that you should read the labels of hair gels and sprays, mousse, fragrances, shampoos, conditioners, body washes, and body lotions to be sure they don’t contain any ingredients that you already know will trigger an allergic reaction in your skin. And if you aren’t aware of any allergies you may have, but you find that you are breaking out or suffering from hives or dry patches on your skin, look to beauty products other than your skincare products as the possible culprits. In particular, eliminate products that contain dyes and fragrances as much as possible. These ingredients add nothing to the efficacy of the product and are often the cause of skin conditions you’d rather avoid.