Does inertia increase with speed? I have heard that when the speed of the object increase, the mass of the object also increase. (Why does an object with higher speed gain more (relativistic) mass?)
So inertia which is related to mass, increase with speed?
So, if I accelerate on a bus, my mass will increase and my inertia will increase for a while on the bus, until the bus stops?
 A: For both interpretations, the answer is 'yes' since force still acts in an opposite force on anything which has mass. As you accelerate, your velocity increases and therefore mass will increase. The increase in mass will bring about an opposite force. The greater the mass, the greater the inertia.
A: It depends on how you define inertia. http://en.wikipedia.org/wiki/Inertia#Interpretations. The answer to your question is 'yes' if we interpret inertia as p=mv, and no if we interpret it as F=ma.
A: I think It has more to do with acceleration than speed. What do you compare a constant speed to without knowing the inertial frame for the universe. For instance if the earth is traveling toward the constellation Leo at 390 km/s what would happen if you blasted off in a rocket in the opposite? After accelerating would you be moving 390 km/s or would you be sitting still compared to what you were doing?
A: I think inertia doesn't depend on speed, it depends on rate of change in speed, i.e. acceleration.  The higher you accelerate the more will be the inertia.
It can be understood by taking an example of a motorcycle, in which lower gear gives more traction than the higher one.  The higher the acceleration you want the more traction is required due to inertia. 
If there is no acceleration then no inertia will be there.
