Total War's AI is programmed to keep you at war. They're heavily weighted to attack you if you have no current wars. Improving relations can help but ultimately if youre at peace with everyone and had 200 relation with everyone... A faction would back stab you.

About mid campaign Carthage and Rome will go to war , its inevitable. A main part of playing Rome is that conflict, and preparing for it. I believe there is a RNG factor also regarding this. Also with diplomacy , you need to look at who likes and dislikes who when making deals and declaring wars.

You want to take out Epeiros, Carthage starts at war with them too and they are your main initial threat.(do not peace out with them that will make it worse)

So your initial targets are Epeiros and Etruria, you want both of these.

Attacking factions Carthage has deals with might be a bad move for relations. Even trade deals with factions at war with them impacts this. Look carefully at their likes and dislikes if you want to prolong the peace. Attacking factions they don't like improves relations, converse is true (same with agreements)

Also keep a small army in south Italy, but don't cross the border. Army placement has an impact on factions attacking you and making deals with you.

My suggestion make a trade deal with veneti, non agression pact, military access and demand money (that will help protect that border, and give you $$) I also like to make deals with Massalia.

You need to take care picking the best territories to own. If they don't have good earnings or strategic position they just add to the empire maintenance,.
This is dated but still has relevant info.

Province Management & Economy Guide [www.twcenter.net]

TIP:
You can find out what deals a faction has quite easy, open diplomacy with them , (look at all the "break X agreements" options for that faction. You can also click a faction here and mouse over another faction on map to get a general feel for their likes and dislikes.

Carthage declaring seems pretty closely tied to either them conquering all of sicily (including clienting Syracuse), but with some RNG amount of turns after that. Roman armies entering Sicily triggers it too.

And as Stix was touching on, diplomacy is a huge factor in improving your position in the world. A LOT of time actually can be spend just investigation each faction to figure out who to be friends and enemies with, then contacting them to make the deals, sowing divide between them creating the axis of good (you and your friends) and the axis of evil (your enemies). Pretty time consuming, but to me at least super relaxing down time between the battles :)

Same here, I try to avoid war just to build up a bit but two turns after taking southern Italy they always declare war (with no armies in Sicily, and Syracuse still independent)