To start, you have to understand nearly everything we know about the Regal period is myth. There are a few people that think they know things about the period: Thomsen published King Servius Tullius that argued that Servius Tullius was real, and discusses his policies. Ogilvie gave it a decent review in The Classical Review, though he says T. ignores archeology. There's an Italian dude whose name I always forget who thinks Romulus was historical and thinks he found Rome's original walls and the House of Romulus and such things. I don't know anyone that takes him seriously, but I don't hang out with Italian archaeologists.

The more sober approach might best be accessible in Cornell's The Beginnings of Rome. It's not my favorite book, but it does, in a text-book sort of way, put forth the archaeological evidence for an early Villanova settlement, or collection of settlements in the hills that became Rome. The archeology gives a story that is very different from the literary tradition.

To return to the literary tradition, Livy is our most complete narrative of the Regal period. As far as I know there are no large gaps in the first book - the bit that deals with the kings. Livy, it is thought, relied on a tradition of annalists - that is, people who kept records of events in a year-by-year fashion, probably associated with temples. From time to time in Livy you find a list of religious portends and such things, and the assumption is that scribes recorded the major events in a year for temples, and those records were preserved, and Livy used them. That's a lot of assumptions, but it's hard to imagine where else he'd get his catalogs of supernatural events. I know a prof at Boulder working on a new Early Rome textbook that treats Livy as sort of the surviving exemplar of the annalist tradition.

A second tradition for Rome's early history comes through Rome's first Latin historian, Cato the Elder. There are about 40 pages of fragments of Cato's history of Rome left. The bits that survive largely survive because Cato was the source material for Virgil, and so the largest chunk of surviving text deals with Aeneas, which is before the founding of Rome, but Cato's history continued through Punic 2, and we have bits and pieces. More importantly, other Romans had Cato and used him. Cicero, in De Rei Publica, provided what I think is a too-often overlooked history of Rome's Regal period. He sets this history in the mouth of Scipio Aemilianus. This character says he's reporting Roman history (and he uses Cato's title, Origines) in the very words of Cato. There's some argument over what this means. If I remember correctly, Cornell left this out of his recent edition of the Fragments of Latin Historians because he didn't believe the attribution to Cato was secure enough, but Astin thinks Cicero dropping the title Origines in there is sort of slapping the reader in the face with Cicero's source. There's another argument that what Scipio implies isn't that he read the Origines, but that he learned his Roman history from discussion with Cato, and it's not at all clear that what Cato said and what he wrote are the same thing. I agree with Astin - I think dropping that title is a pretty hard indicator of what Cicero was using, but that isn't going to convince everyone. Cato also was an important source for Dionysius of Halicarnassus, whose early books are in reasonable shape, though there are big gaps.

A third tradition comes out of Rome's greatest scholar, Varro. Varro was a first-century BCE polymath, and among his many other works was a Roman history. Here is what the New Pauly has to say about it:

Some 70 fragments survive. The subject is historical research, certainly also including speculations and etymological derivations of Roman civilization, with the aim of constituting an independent, tradition-related consciousness of the Roman world, to free it from a customary conception of cultural dependence on Greece.

Varro's work was the basis of the Fasti Capitolini, and his chronology differs from Livy's. I wish we knew more about his work. He was very influential on people like Augustus and Cicero.

So there are at least three major sources for traditions about the Regal period. Livy's is the best known and the most in-tact, but I'm not sure you could say he's the most important. For one, if you really want to know Roman history you have to look at archeology. Second, the tradition of Cato was as important as the tradition of the annalists. Third, Varro's book it seems had a larger impact on its time than Livy's did; Livy happened to survive.