To defeat the Hellenistic armies you oughtn't try to fight their phalanxes, but, as Morte66 explained, try to manoeuver and break them. But don't risk your weaker Caucasian troops to kill them, you'll need better infantery to dare that; later Armenians or Babylonian mercenaries of all sorts.
Slingers are very useful, if you're used to their range and ballistics, else they can be a pain, cause you can't put them directly behind your lines. They have to manoeuver swiftly, in order to find clean fields to shoot, but they shoot hard; you'll have to have them supported by threats from general's bodyguards or other cavalery. Eastern archers can do the safer job behind the lines, but you shouldn't forget to be very swift, you cannot stand ground if the phalanxes move. I personally find that foot and horse skirmishers are very useful, it doesn't matter how many die, they can be retrained in almost any town you'll possess.
Horse skirmishers are also valuable, and they can kill off the fleeing units afterwards (this is important)

If you aren't used to eastern non-hellenistic warfare it's a bit hard to begin, you can't fight like the Hellenes, Romans or Celts and Germans, cause in the first years you won't have a lot infantry that can stand in a fair fight against enemies. Most of the spearmen are for garrisons, missile troops don't stand on themselves, and it's only after you got the heavy blows that your troop options will ameliorate You should indeed hold on to your mountainions properties until you have better armies, then come out of the mountains and conquer, so now it's best to have a stable basis in the Caucasus (and don't try to 'consolidate' by having only Homeland; look for troop diversity)

BTW, Kartvelebi Dashna-Mebrdzolebi rock