I got Rome: Total War and I also have Barbarian Invasion and Medieval II: Total War, so you could call me a fan of the game series. Best faction is Macedonians or Seleucid Empire. Both have good phalanx units and both do have slingers
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Seleucids also have elephants that cut huge holes into enemy ranks and cause terror. The only problem with Seleucids is their campaing location, Egyptians in the south, Parthians in the east and north and Pontus in the west. Those factions also tend to be rather hostile towards the Seleucid Empire and it is usually the first one wiped off from the face of the earth.
Macedonians are rather easy to start with. Due to their extra long spears the Greeks are a quite easy to beat. You can also launch an attack towards the Thracians. They don't really have a chance because they are very weak faction. You can usually get assistance from the Dacians. Your worst enemies will be the house of Brutii and Dacia.
After all it's a good game but it is rather old and I have played it through several times so I'm getting quite bored playing it. I suggest that you buy Medieval II: Total War. It needs a quite good computer but it is worth it.