The Greek Cities are among the quirkiest and most challenging factions in Rome: Total War. They're also one of the most fun. They are, however, daunting to an inexperienced player, because their territories are so far-flung and they share borders with so many powerful enemies, including Carthage, Rome, Macedon, and Seleucia. They actually own only two mainland Greek cities- Sparta and Thermon- along with Rhodes, Syracuse (in southern Sicily) and a single city on the coast of Asia Minor. When I first attempted the Greek campaign, I found myself overwhelmed by the sheer complexity of micromanaging the Greek Cities. Because every individual Greek city is completely isolated from the rest of the faction, they must maintain a very high degree of autonomy and self-reliance, whereas centralized factions can afford to specialize their cities for certain economic or military activities. Not to mention, their lack of heavy or even medium cavalry puts them at a major disadvantage against their most immediate enemy, Macedon, as well as most every other faction in the Mediterranean. Their hoplites also carry shorter spears than their Macedonian and Seluecid counterparts. Fortunately, they also have an interesting set of geographical and tactical advantages that make for a challenging but totally playable experience.
First things first, you've gotta understand the differences between the Greek armies and the Macedonian/Seleucid armies, which are their main rivals.
The Greek armies are made up of these units (units on the right require more advanced structures):
Militia Hoplites-> Hoplites-> Armored Hoplites-> Spartan Hoplites
Peltasts-> Archers-> Heavy Peltasts
Militia Cavalry-> Greek Cavalry
The Macedonian and Seleucid armies are like this:
Militia Hoplites-> Levy Pikemen-> Phalanx Pikemen-> Royal Pikemen (Mac)/ Silver Shield Pikemen (Seu)
Peltasts-> Archers
Light Lancers (Mac)/ Militia Cavalry (Seu)-> Greek Cavalry-> Macedonian Cavalry (Mac)/ Cataphracts (Seu)-> Companion Cavalry
The Seleucids also get Elephants, War Elephants, and Armored Elephants.
Greek Cavalry is simply inferior to Successor State cavalry. Greek Cavalry is pure light cavalry- they are fast, light, and relatively cheap for horsemen, but the downside is that they get torn to pieces in any kind of heavy combat. They don't have a prayer in battle with heavy cavalry. Militia Cavalry are even weaker, but have the advantage of being able to throw javelins, even while moving. I generally prefer them over Greek Cavalry in most cases, because when faced with enemy cavalry, they can simply skirmish and throw javelins as they fall back, thereby denying the enemy battle, dealing casualties, and serving the ultimate purpose of keeping the enemy cavalry away from my phalanxes' flanks. They are also useful in a supporting role, because they are fast and strong enough to run down enemy missile troops, or to hurl javelins into the flanks of enemy heavy infantry.
Greek missile troops are exactly the same as Macedonian and Seleucid troops, exept for a unit that is totally unique to the Greeks: the Heavy Peltast. These javelin-men are also pretty solid light infantry, well able to beat peasants and militia troops. Their large shields and overall maneuverability make them a great screen for slow-moving Greek phalanxes, and once the fight starts, their superior mobility makes them useful flankers. They are hardly a game-winning unit, but they are one of the precious few advantages that make the Greek Cities playable.
Where the Greeks really shine, though, is infantry. Hoplites are outstanding troops, among the best in their tier (cost-wise). Statistically, they're very similar to Rome's Hastati or Carthage's Libyan Spearmen, but their ability to form a phalanx, coupled with their superior stamina and morale, make them more than a match for either of them in a frontal attack. They're also fairly handy swordsmen. This sets them apart from the Levy Pikemen, the Successor infantry of the same technological tier. Levies are basically unarmored, clothed only in a long tunic, and their shields are substantially smaller. Their only advantage is the length of their spears, hence the name Pikemen. The same is basically true of Armored Hoplites and Phalanx Pikemen; Phalanx pikemen wear light leather cuirasses, whereas Armored Hoplites wear the full, old-fashioned bronze panoply, and are crack troops to boot. Armored hoplites are far superior to Rome's Principes and Carthage's Libyan Spearmen. Spartan Hoplites are the best regular infantry in the game; although they have low armor ratings for a phalanx unit, they have absurdly high attack for hoplites and also sport excellent stamina and morale. Royal and Silver Shield Pikemen are armed exactly like Phalanx Pikemen, they just have slightly better morale and fighting ability.
So, this creates a unique tactical situation. In a straightforward infantry battle, the Greeks are basically peerless. This means that Syracuse, threatened by the infantry-heavy Romans and the lightly-armed Carthaginians, is quite secure as long as no mistakes are made. Rhodes, well away from any enemies, is also fairly secure, and makes a great naval base. It is in Greece, therefore, that the real danger lies. Most of Greece is occupied by Macedon- Thermon and Sparta are under the control of the Greek Cities, and Athens is currently independent, but will invariably fall to the Macedonians in due time. Geographically speaking, the forces in Thermon need only hold the line as the Spartan soldiers drive North. It seems easy enough on the map, but the real challenge is in actually winning those battles.
Macedon has excellent cavalry, and will begin taking advantage of it immediately. The Macedonian phalanx may be frail compared to the Greek one, but it isn't designed to kill Greeks; it's designed to hold Greeks in place while the cavalry kill them. Because you can't hope to match them in cavalry battle, except through sheer weight of numbers, your best bet is simply using large hoplite reserves to repel flanking attacks, in concert with a little light cavalry. Archers are useful for blooding enemy cavalry from a distance, and peltasts (javelin throwers) are extremely lethal against the Macedonian phalanxes, with their pitifully poor armor. Keep in mind, any cavalry used in battle with the Macedonians is used more for pursuit after a rout than for actual combat, because they will suffer catastrophic losses if they engage the enemy horsemen, but it's worth having them. You basically have to rely on sheer muscle and killing power to break the Macedonians- their system is tactically superior to yours. Therefore, you want to get as much bang for your buck as possible, by making each battle as bloody for them as you can. That's true of just about every battle in this game, but even more so for the Greeks, because their economic and geographical underdog status forces them to be miserly with both money and men.
Upon clearing out Macedon, you'll have a series of new threats; the barbarian kingdoms Dacia and Thrace, and, more likely than not, the Roman Brutii clan as well. The Seleucids of Syria and the Pontic kingdom of Asia Minor are, doubtless, also threats by now. Fleets from Egypt will probably be in your waters as well. The possibilities are endless from there.
EDIT: I decided to add a picture so that you have a visual aid. This is of Militia Hoplites, apparently Macedonian (judging by the pattern on their shields), but their spears and shields are a good representation of the Greek hoplite infantry in general. The pikemen's spears are long enough for 5 spear points to extend in front of the ranks rather than three, but their shields are very small and strap onto their forearms (i.e., they're pretty worthless).
Normally, these Roman Hastati would butcher the Militia Hoplites, especially since they have the advantage of charging downhill. However, the spearwall created by the phalanx formation holds them at bay and gives the otherwise-outmanned hoplites a fighting chance.