I'm pretty sure doing what the senate asks and continually completing the missions they set you makes them happier with you, over time.

But don't forget-the only way to win the game involves making the Senate rather angry, i.e. taking Rome.

To make the People happy (and so to get their support) you must expand and build a large empire, to make them look on you as a hero and mighty general. As far as I know, building things and throwing Games does not make the People happier; and the 'people' are the plebs of Rome, so they aren't affected by your empire directly. If you have around 25 provinces and haven't lost too many battles the Plebs will probably give you support to attack Rome.