Hello and welcome!
Oils that make your soap lather are: Coconut, Babassu, Palm Kernel (Use at no more than 20% - or only 10% if you want it to be less harsh)
Oils that help with hardness/longevity are: Lard, Tallow, Palm, Shea Butter, Cocoa Butter or Soy wax (Use a combination of these up to a...