The Soul Puppet will need to meet the minimum required level to unlock each succesive skill. For example, a Succubus can learn Sharp Shot at level 1, but cannot learn Moon Pierce until it reaches level 10.
There are two ways for normal Soul Puppets to learn their first five natural skills; the first is by visiting the Skill Shop and purchasing Skill Items, and the second is by hunting in certain dungeons until they drop.
You can view a full list of skill items, along with the creatures who use them, by clicking here.
Some creatures, such as Credit Shop purchasable guests, customs, enemies and Limited Edition Soul Puppets, can learn their skills from within their status menu as they reach the appropriate level for each ability.