Hmm, for me, when I play as Carthage or Seleucids, elephant generals save the day in the early campaign while the faction cannot afford sturdier regular troops.
I engage the enemy with cheap infantry and then charge the elephants into the rear or flank (preferable one flank) of the engaged enemy. Do not let the elephants get bogged down in melee in one place. Charge in and then just follow through charging the next unit, then the next. Due to their huge mass, elephants will just push through resulting in a mass route in no time. This works best when elephants charge into a flank of an extended enemy line and then just continue charging through the whole front.
In field battles you have to have some cavalry to support the elephants though (light cav is usually sufficient). To be more exact: cavalry needs to keep AI's skirmishers away from your elephants. Pachyderms are extremely susceptible to missile fire (especially fire missiles). Beware of missile cavalry: those can spell death to elephants, especially if handled by a human player.
I am not qualified to comment on chariots: I rarely use them myself.
p.s. In Carthage's campaigns, I use elephant generals to a great effect in walled city defense battles. I hold the area by the gate with cheap hoplites (garrison units usually) in phalanx formation. I allow the AI to engage the phalanx. Wait a bit for exhaustion to lower the morale of attackers and then charge elephants into the rear of the blob. A mass rout usually almost instantly. Again, you have to beware of the AI skirmishers on the other side of the gate: those can kill elephants or force them into a frenzy.
Elephants are great anti-heavy cavalry units as well but only if they catch a cavalry unit in one-on-one battle without other AI's units joining in. In the latter case, elephants die easily.
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