Which particular transistor types represent triode and pentode vacuum tubes best?

None of the above. They are voltage-controlled, like FETs, but their characteristics are not so similar. Perhaps JFETs are closest, since a more negative voltage on the grid turns them 'off'.

You should just look at the I-V curves of vacuum tubes for a basic understanding.

Consider that if the anode current is fixed at 1.5mA and the grid voltage goes from -1V to -2V, the anode voltage goes from 160V to 260V - a gain of -100!