Carthage might be hard to play as but can be very rewarding. The are able to recruit the elephants and their cavalry and infantry is also impressive. However, they do suffer from very poor early game infantry, meaning they must rely on cavalry to win early battles. here is somethings i done during my campaign early on to ensure later success:
1. prepare to abandon corduba to spain if attacked.
2. give caralis to the Greeks as a gift.
3. train a strong navy.
4. conquer all of Sicily
4. destroy Numidia.

Now lets looks at their potenial enemies:
House of Scipii.
basic plot is simple, the scipii will attack Greece first, doesn't matter. using your faction leader with his force and attack messina on turn2 (recuit mecenary hoplite). the scipii force besieging Sycrecuse will turn around and attack you. defeat them in battle and messina is yours. the scipii won't take this nicely and will bombard you with forces from Capua. The solution is simple. train up a strong navy. and destroy their force will at sea. mainland Italy is separte from Sicily. the scipii will need to use their navy to make landings. don't let them! sink their ships and their force on board will sink with them. keeping on doing this and the scipii will not be a worry any more. you'll conquer Sycrecuse later and rule all of Sicily.

Numidia
These guys will come and ask for treat first and sneak up on you later. Carthage will never be safe with Numidia ready to backstab any minute. the solution is simple. destroy them, and as quick as possible. you'll have 1 family member in Palma and Corduba. move them out and recruit regional mecenaries. belaric slingers and Spanish Mercenary will best any infantry you can train right now. move them out and attack Cirta. (remember to ask map information first. Numidia is a awfully big comparing it's land mass). after that take Tingi and Dimidi acordingly. Take Nepte as well if it's held by the Numdians. After that they'll only have Siwa left and that is separated by Lipcus Magna. don't bother with them any more. you rule all of Western Africa without any worry of being backstabed. all of you African cities is safe, for now.

Spain
The chances are they'll try to take Corduba, but if you're lucky, like me. The Spanish will offer alliance and actually keep it. this is idea sitiuation. If they do decide to attack Corduba then just let it go. you'll get them back later.

Julii
Sooner or later the Julii will attack Caralis, if hadn't given it to the greeks already then just let the Julii take it without a fight, they'll be content after that and apart from their navy. you won't even notice they're there.

Brutii
Brutii might ocasionally block the port of Syrecuse. just make sure they don't land any troops in your region. most of the time the Brutii won't take aggresion against you until you take aggression on them.

After 40 years you'll be ready to take on Rome. The city of Carthage is growing fast and will be your troop production centre. Thapsus and Lilybauem will also reached large city at least. your core infantry should be made of Poeni Infantry with Sacred Band Infantry as elites. use Sacred band Cavalry as shock cavalry. Longshield cavalry haven't lost all it's use and will make it to your cavalry selection because of it cheaper cost compare to Sacred band. Use Elephant as weapon of mass destruction. early on these guys will your key to victory. i tend not to use them too much later during the game. in my case i ignored missile troops although. since Punic missile troop sucks anyways. it is important to wipe out all Roman factions before the Marian reformation, although it is more fun fight Legionaires. In my campaign i took Sicily first then conquered all out Western Africa. i didn't take Lipcus Magma because I wanted it to act as a buffer between Carthage and the remaining Numidian city. Corduba was untouched and Spain and Gaul fought over Celtiberia. Caralis was taken over by the Julii. 237BC i was ready. my army rolled into Italy. Rome was the first one to go then followed by everyone else. Brutii still had lands in the Balkans. I didn't care. I should all their navy and they had attack by land. my next step was the Iberian Penisular. the spanish had chased the Gauls out. luckily by that time Corduba already reached Large city and could look after itself. Spain attack Corduba, they were beaten back. a counter attack followed. the Spanish wars last for 15 years before the factions being totally destroyed. Gaul was a consistant pest on the northern borders but i had no intention of going into northern Europe. way too poor. next i journeyed across Africa and destroyed Numidia. then i was draged into a war with Egypt. this was the most fun part of the game. Egypt destroyed the Seleucid already and was the dominated faction of the east, till i came along. the long Egyptian-Punic wars lasted for more than 5 decades before Egypted was totally crippled, not destroyed. after fighting Pontus and taking Anatolia i was done and completed my grand campaign.