I visualise it as a classic mass-spring-damper system (a spring and damper in parallel, solidly mounted at one end with a mass attached at the other).
The proportional element is equivalent to the spring - the more deflection (error) there is, the harder this element will try to push the system in the opposite direction.
The derivative element is equivalent to the damper - the faster the deflection is increasing, the harder it will try to oppose it; the faster the deflection is decreasing, the harder it will try to maintain it.
The integral element is like a ride-height controller - if the system is held away from the desired position then it will slowly try to push it back towards the desired position.
When tuning the PID controller, I suggest you tackle the elements in that order: you need to have the proportional element about right to give the system enough authority to correct deflections. On its own this will probably be unstable (unless the system you're controlling is inherently stable) since the correction always lags the error slightly to oscillations will tend to be amplified. Then add damping until the system does not spontaneously oscillate or overshoot - you want the minimum damping necessary to achieve that. Then add the integral element to deal with any tendency to drift or settle away from the desired position.
This approach addresses one degree of freedom. I'd try to deal with each one independently in turn if possible, but you may find that there is cross-talk between them that has to be addressed when you try to combine them - but getting each DoF controlled in isolation is a starting point.