Hey Niroc,
first of all, i play with the 4-turns-per-year and the supply-submod.

Rome has a really good start into the campaign, because you got some good provinces right from the start. But you got some crappy buildings in them. Roads for example are a waste of money.

So for me i start with the battle of Zama of course, then capture Palma with my spanish army right in the first round. After that i build 3 provinces with economic buildings: Italia, Magna Graecia and the islands (Corisca, Palma and so on). The first two i build up as farm lands. The islands are perfect for a trading province.
I ignor all my other provinces until i got the money to build them up too.

Key is to specialize the province for one economic purpose. For example, when i build a farm province, then every building in this province is a farm or boost farming.

When a city got a trading good then this building is a must. Trading goods are extremly good.

I put some Screenshots of my provinces in my current rome game on my steam profile, you can watch them there.

Hope this helps a bit :)
Greetings
FreeParmenion

send out a ship to find as many people to trade with from turn one.. build as many good resources up first .. gold silver ect .. and trade and like the comment said above farms are good to get food and money build them up .. after awhile you will have to much money i as rome make like 80000 a turn profit .. so its needs some balanceing still but if you can balance and again like the above comment said get rid of wasteful buildings you will be just fine

Thanks for the tips guys. Any key provnices i should really focus on aside from italia or?

spain has lots of resources .. gold bronze silver and if you can its easy to make money from trade with the other factions there while you deal with carthage

I rushed colosseums, public baths, and wells, as well as the resource buildings E.G olives, fish, etc.

Traded with as many people as possible.

I abandoned a couple of cities / regions in spain, due to the public order being so low early, that you pay upkeep on buildings but get zero tax, and it gives you negatives.

I used the granarys or whatever, for the set amount of food, as in winter your food drops massively.

I kinda just focused on defending while my latin influence spread from my buildings, and public order stabalised.