It's actually easy to play as Rome if you know a bit of history.
Carthage at the start of the game is pretty weak, and seems to be at war with everyone--or be at war with everyone eventually. You should declare war on it within 2 turns, and start taking the islands with your preatorians.
After this send an army to northern africa, while ignoring Greece and Egypt. 8 preatorians and 3 trebuckets is MORE than enough to steamroll through northern africa--and add in a great general too, if you want to go even quicker.
While you're destroying carthage's northern africa, build up your infastructor so you can become a war-machine quickly, and then start massing an army to battle with Greece. You'll need a much larger army this time--I went with 15 pratorians and 7 trebuckets.
Oh yeah, a tip. When taking northern africa, don't leave praetorians behind to guard the cities you don't destroy. Just use slavery to pump out an archer for each city.
Anyways, eventually when you take all of Carthage's northern African cities, the country will usually always ask to become your vessal for life. You might as well agree, since there's no use in taking the rest of his cities in spain.
Build up your northern african army a bit, and then sail them towards Egypt. While doing this, finish up with your Greek army, and then march it towards greece. Egypt is almost always weak as heck by time I get to it, from a mixture of barbarians and not being the best at managing their empire, so I don't have to worry much there. Greece can be awfully strong though...
Oh yeah, and build up a good sized army on the side, along with a nice navy, to take the islands of Greece. Try and get all the wonder cities.
Get Rome into 'war-machine' phase, and then declare war on Greece and Egypt at about the same time. Hope for the best. I usually force both of them to become vessals, since it's useful not having to deal with such a massive empire. After this it's as simple as moving your armies back to northern rome, and then plowing into the Celt's stupidly strong nation. Just destroy the celts completely, except for London and a few cities in France.
About having Greece as a vessal. I've always had some trouble with them as this, and sometimes I've thought it better to just conquer them. They seem to like spreading northward, and then building up a massive army. Unless they make themselves vessals for life, it's risky. Greece will actually work at overpowering you if you're not careful. Carthage doesn't though, since they lack room. Egypt will just sit there.