I've finished about 5 campaigns as Carthage and the best way to a great start is
to move your army on Sicily to the Scipii city and take it the second turn, it really isn't that hard, especially if you hire the mercenaries, after you take the city the Scipii are screwed, finish off the Greeks on Sicily and from there go into Italy.
Early on your army should consist of mostly rs cavalry, Carthage can mass produce rs cavalry right from the start, and in my opinion your main unit early on is rs cavalry not iberian infantry, massed rs are very effective against the romans and their much cheaper than your infantry.