OK, as a guy in Hayasadan campaign right now, I have a general action guideline for you.

1. Disband the cavalry at start and take Kotais (Phraaspa is a long walk and will leave you pretty far in debt. Use the leftover troops to take Trapezous.

2. Build (roads-800 mnai 2 turns) and (garrison/city defenders-2000 mnai 6 turns) in Armavir. Meanwhile, raise a small army to seize the town of Kabalaka, to the northeast. DO NOT build new governments in the new provinces.

3. Once you've built the aformentioned buildings, Homeland Resource should appear in Hayasadan/Armavir. Destroy the (Lv 3 government/Marz/Satrapy) and build a (Lv 1 Government-10 turns 2400 mnai).

4. Once that's done, better homeland/subjugation resources will appear in the provinces you've conquered. Build Lv 2 governments in those provinces.

5. Save up money to build Hanq/Mines in Kotais. This will get your weak economy moving, with 2 mining towns.

From there, you should have sufficient ability to take the Caucasus. See the EB FAQ if this wasn't clear.

--- The Most Straightforward Unit Guide Available for the Early Turns as Hayasadan ---

Kavakaza Sparabara - "Caucasian Spearmen" these will be the backbone of your early armies, and much of what you'll use to conquer the Caucasus. They're the best of the cheap - DO NOT BUILD PANTODAPOI (crappy spearmen of death). These guys are all-purpose guys early on, as they have large numbers for garrison and long spears for field fights, etc. but should ONLY be used in a field fight if the enemy have been softened by missiles first. They are part of the "Hayasadan Triad"

Kovkasi Lernain Netadzik - "Caucasian Archers" these are amazing. For a cost of ~600, you get 6 missile attack at a formidable range. They dominate Caucasian warfare, and are the only reason the Sparabara are considered decent in your armies. They decimate the ranks of lightly armoured troops, the majority of what you will fight, even from the Seleukids. These will be used even when you have good heavy infantry and armoured HAs. They are more than a support arm: they are your main arm in battle. Only reach to the Kavakaza Sparabara after these guys have emptied their quivers (unless you're forced to), for otherwise the Sparabara will be defeated.

Khuveshavagane Shahvar - "Royal Kinsmen" they are how you will win your field battles, and they clean up sieges along with the spearmen after the enemy routs to the town square. On the battlefield, you will employ an Alexandrian system: the Netadzik will advance to empty their quivers, then the Kavakaza Sparabara will advance to fight the enemy, while these guys wheel around behind for a devastating charge which will almost always rout them. Try to always pull the charge from behind quickly so the Sparabara won't take too much damage, but don't move the cavalry in place too early, as the enemy will run after them and that's just annoying.

--- Potentially Useful Auxiliaries ----

Gund-i Palta - "Peltasts of the East" they are mainly used to volley into enemies' sides during either siege or pitched battle, but they can also be used to swarm enemies after volleying in order to rout them via flanking and/or numbers.

Shuban-i Fradakhshana - "Eastern Slingers" they are slightly worse archers against lightly armoured troops, but vastly more effective versus armoured units. Use them mainly as specialists vs. armour, but don't feel bad using them against unarmoured units as well.

Kartveli Dashnit Meombrebi - "Georgian Medium Infantry" they are excellent replacements for Sparabara when you can get them, but require a Lv3 MIC in either Kotais or Mtskheta.

Hai Nizagamartik - "Armenian Spearmen" they are narrowly better than the Sparabara, but also have fewer numbers, meaning worse garrison but less of the common "pop drain" due to necessary over-recruitment. Mix them up with the Sparabara, but don't rely on them solely.