First things first, nanobot skills are the placed in the keyboard slots Q,W,E,R and Z,X,C,V, they are the "special" attacks of whichever nanobot you have equipped. However, the new skill system requires one to fulfill certain requirements in order to learn new skills. Each nanobot (including evolutions) has the potential to learn a skill that you can use for all nanobots of that type (ie. Knife (blade), Attach (fist), Sword, and Dual Wield).
Also, some nanobots will grant higher ranks of skills, such as lvl 2 and lvl 3 skills from some of the nanobots later on.
The most common condition up to the lvl 19 nanobots depends on mob kills with that nanobot, after a certain number of kills has been reached you learn/unlock that skill (press "g" to pull up nanobot information and the skill is near the bottom).
After lvl 19 however, most of the nanobots require reaching a certain level with that nanobot in order to learn the skill.
Special Thanks to paifrenzy on the GXU forums for making a list of most of the skills