There's a "native" barracks line for each faction, and a "foreign" barracks line for each faction. The native one gets you the Roman legions (or the troops for whatever faction that you're playing) we all know and love. The farther away you get from your homeland, the fewer (and less elite) troop types this barracks line will give you. The foreign barracks gives you troops similar to the people you just conquered. For instance if you build up the "foreign" line in Athens you can get Greek hoplites. If you build up the foreign line in barbarian areas, you can usually get the lower level type of that troop... but foreign barracks rarely give you anything in your homelands. Rome's a little wonky because which line of barracks gives you the most benefit changes as the game progresses (see the EB FAQ about Reforms - but don't worry, you can conquer the world even if you don't do all the reforms).
Its hard to tell these apart when you're new to EB (and harder when you're trying to figure out which to destroy. Read the description of the buildings to figure out which line of buildings is foreign and native (I don't feel like looking it up for Rome, should be fairly obvious). When you take a city, if you look at a barrack building's description, scroll all the way to the bottom and it will tell you what faction can use it. Occasionally it's you! If you play as the free Greek cities (Koinon Hellenon), for instance, there are a lot of rebel cities with a Greek culture that come with buildings ready-made for you.
As far as culture, when I take a city I intend to keep for a long time, I frequently destroy almost everything I can of an opposing culture (marked in parentheses after the building, unless you built it, then there's nothing - Rome's is Roman). Sure that trader building gives you more monies, but you can always rebuild it and its causing your people to be unhappy because its another culture.
Anyway, look at the sticky threads for introduction to EB mechanics, etc. It's a great mod, you should enjoy it even if you don't enjoy history. If you do, it's a dream :).
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