Here are some broad strokes. By the end of the regal period (end of the 6th century BCE), Rome was an expanding city-state with a tripartite political system: the Senate, a popular assembly, and an executive (kings, and then consuls). We are told the inhabitants were grouped into thirty units, curiae, which divided into three tribal groups of ten. These divisions formed the basis for political structure (the most ancient assembly of the People, the comitia curiata), and also provided structure for military recruitment and organization.

There was a well-established Iron Age aristocracy, which was virtually identical in its behavior to other Indo-European Iron Age aristocracies all over the Med. They were buried in graves loaded with rich grave goods like weapons and exotic imports; they dwelt in elaborate houses; and they organized themselves into familiar cabals which eventually probably coalesced as the well-known Senate.

We don't know for sure, but it seems that Rome, though just of several emerging powers in the plains of Latium, was already a little bigger and a little more aggressive than them. Places like Tibur, Praeneste, Ardea, and Lanuvium would soon find themselves in the way of Roman expansion in its most infant stage. The city was a slightly bigger fish in a relatively small pond, and not yet ready to come up against the powerful Etruscan cities to the North, or the Greek colonies to the South.

Culturally, the Romans belonged to a larger "Latin" or central Tyrrhennian milieu. There was a lot of linguistic, artistic, architectural, and material borrowing, including with the Etruscans. The Romans' first contact with specifically "Greek" things came via an Etruscan medium. This included some early gods, like Herakles. Besides these Etrusco-Greek imported deities, the Romans worshiped a whole pantheon of Indo-European (Italic) gods pertaining to fertility, the sky, etc.