PID Parameters effect while tuning
In practice it is always beneficial to have some kind of model for your object. One useful mathematical model is transfer function. It is hard to get the ideal one but fortunately most of the objects can be approximate by first order inertia. 
$$
G\left ( s \right ) = \frac{k}{Ts+1}\cdot e^{T_{0}\cdot s}
$$
Where k is gain, T is time of inertia and T0 is transport delay.
Here is some simple method System Identification Using Time Domain Data
Having such model you can simulate your object with different controller's loopback algorithm using Matlab or Simulink. 

In most cases you are not allowed to experiment with the real system but if you can then try this: choose P action first, test the behaviour of the real object then correct it test again and so on. Then you can add I action, test the behaviour of the object then correct it test again and so on. Then you can add D action, test the behaviour of the object then correct it test again and so on. Remember D action is hard to control and it is not used so often as it may seem. If you want to have steady state error = 0 always include I action (it is obligatory to achieve error = 0 in practice).

Your general rules are right but the issue is not so easy. PID's parameters highly depends on the process you want to controll and stability of the system is a different story. Search for example for Hurwitz stability criterion.