The regal period largely coincides with the Archaic period, for Rome's development.
Of course, a lot was going in what would develop into Rome before the famed foundation date of 753 BC (which was debated by ancient historians nevertheless).
The 8th and 7th centuries are known as the Iron Age and Orientalizing period and were marked by shared and overlapping traditions and practices up and down Italy, involving Latium, Etruria, and Magna Graecia.
For the Romans' own consideration of the early city, a quintessential starting point is Livy, whose first five books begin with Romulus (and earlier still, the tradition of Evander and his fellow Greeks).
Some examples of monuments and projects attributed to the early kings:
Romulus - Temple of Jupiter Feretrius on the Capitoline hill
Numa Pompilius - Temple of Janus in the Roman Forum
Tullius Hostilius - Curia Hostilia (first Senate house)
Ancus Marcius -- fortification of the Janiculum, Pons Sublicius
Tarquinus Priscus - terracing of the Capitoline hill, draining of the Forum area
Servius Tullius - first extensive wall circuit of the city, Temple of Diana on the Aventine hill
Tarqunius Superbus - Temple of Jupiter Optimus Maximus on the Capitoline hill
In terms of the archaeology of the early city, the remains are piecemeal, as one would expect, although at times astoundingly impressive.
We can look at early remains of the Temple of Jupiter Optimus Maximus, the Servian Wall, the sacred area of S. Omobono, and the Roman Forum itself. For the Forum, we can note the Lapis Niger, Mundus, Venus Cloacina, Lacus Curtius, and Curia Hostilia. (We will cover the Forum and its monuments in a later episode.) Of course, there are many other areas that have yielded interesting remains, such as the Esquiline cemetery (remains of which are housed in the Centrale Montemartini museum) and most recently on the Quirinal, quite possibly the earliest remains of the Temple of Quirinus.
For further reading:
The Beginnings of Rome: Italy and Rome from the Bronze Age to the Punic Wars (c. 1000-264 BC), by T. J. Cornell
"On the Origins of the Roman Forum" (American Journal of Archaeology 94.4/ 1990), by A. Ammerman