I use a variety of oils, but coconut oil and olive oil are my two main ingredients. I also use shea butter alot, and if I'm making a facial soap, I will use jojoba, macadamia, and avocado oils.
I do use synthetic fragrance oils, but I can make soap without it. Essential oils are usually easier on the skin, but the saponification process soemtimes "eats" the scent and it ends up not smelling like anything all. Lemongrass essential oil is one that I have found that stays intact in the soap. Key Lime essential oil is also one that stays fairly well.
Newver tried peppermint essential oil in soap, although I do have an essential oil called Japanese Peppermint. I know it's kind of boring, but you might be one of those people who does better with soap that has no scent at all.