Q. What are the steps for my Roman general obtain a triumph?

A. The first step on the road back to Rome in triumph is the "Vanquisher of Carthaginians" trait, obtained by winning a battle or two. Next, after more conquest, you'll get "Conqueror of Carthago" (It wouldn't be much of a spoiler to tell you that conquering Kart-hadast itself will get you closer to a triumph than conquering Lilibeo). (similar traits and criteria exist for most of Rome's immediate neighbors).

That settles those requirements that can be easily obtained by your general. Next his men must name him "Imperator" (commander). Preferably, after a numerically important and difficult battle, won decisively. In history, there were instances of a general paying his men to get the title, but we're not going to let you off that easy.

Exact criteria for Imperator trait:
Spoiler Alert, click show to read: 

Number of enemies on the battlefield:
Small unit size: more than 250
Normal size: more than 500
Large size: more than 1000
Huge size: more than 1500

More than 30% of the enemy killed (you want a clear or heroic victory)
Even odds or worse
Attacker or defender, doesn't matter


Finally, after being named "Imperator" by his men, the general must be awarded a triumph by the Senate. In history, they sometimes made men wait or refused to award it, so having some pull (i.e., Influence) helps here
Spoiler Alert, click show to read: 
(more than 4 is preferable, and 8+ is great!)
.

Good luck!

Fighting rebels won't get you a triumph, but a Corono Ovalis is possible there, by winning a difficult battle decisively.