Coconut oil and palm oil are both popular carrier oils for making your own soap. Coconut oil adds lots of lather, and palm oil adds firmness and stability. Both come deodorized, so they don't scent the soap. The best soaps use more than one carrier oil, to balance hardness and lather with moisturizing qualities.
Coconut oil (with a 76-degree melting point) is a popular choice for soap making because it has a long shelf life and is good for your skin. Coconut oil with a higher melt point (92 degrees) is better for use in tropical climates. Fractionated coconut oil is highly moisturizing and is light, nongreasy, and nonstaining; it has an indefinite shelf life. All coconut oils add some hardness to soap but they are mainly used because they lather luxuriously. Soap made with 100 percent coconut oil will lather even in salt water. But coconut oil can be harsh and drying if you use too much in your soap recipe.
Palm oil contributes hardness, a creamy but not bubbly lather, and some moisturizing quality. Although less drying than coconut oil, palm oil can also be drying if you use too much. Use 50 percent or less of palm oil for hand soap and less than 30 percent for facial soap. Palm oil is considered to be a good substitute for tallow for those who want to avoid using animal products. Unrefined palm oil is orange in color and will tint your soap yellow or orange, depending on the percentage you use. If you want a white soap, you need to purchase refined palm oil. Be mindful of the source too; use sustainably farmed palm oil to protect the habitats of orangutans in Borneo, which are threatened by palm oil harvesting.
If you're new to soap making, begin with a basic cold-process recipe. A typical list of ingredients would be 16 oz. of coconut oil, 16 oz. of palm oil, 16 oz. of canola oil, 6.9 oz. of lye, and 15.8 oz. of water. To this you can add essential oils of your choice. Lye, a necessary ingredient in soap, is highly toxic, so always take precautions when handling it. Check out the link in Resources for complete instructions and other recipes.