Robotics Experts: How to do trajectory planning for a robotic arm
One "quick" solution (not necessarily that quick, but quicker and more robust than starting from scratch) is to use something like Blender to generate position information for a configuration you want your robot to take; Basically your input is
two or more states the are should be in, and blender can compute the necessary angles and interpolations between the two. Since Blender is scriptable using Python, you might even be able to drive the robotic arm "real-time" from inside Blender: obligatory Youtube video.

If your interested in how to write code which could do the solving part for you, the key algorithm to understand is inverse kinematics. This is all about finding the joint parameters given a desired position/orientation. It is not something which is easy to solve (it requires solving non-linear systems of equations), but is certainly do-able.

Close-loop feedback is not necessarily required, however you do need to be able to accurately position limbs. One open loop solution is to use stepper motors and a known "home" position. The software can track what it thinks the current orientation of the arm is by simply tracking what operations it did moving away from the home position.