He's a very readable author (in translation I hasten to add) and I'd encourage everyone to pick up a volume. But I imagine that lots of things he records are... of variable accuracy (as is normal with historians of the time). In his "War With Hannibal", for example, he gives supposedly verbatim accounts of all speaches made by all the generals before every battle. But they all appear to speak in the same style that one Titus Livius writes in.
That's clearly a bit of added colour by old Titus. And it's great colour, worth reading. But the battles he describes are well attested with other sources and some archaeological evidence, I understand.
But what of the earlier history? He describes Romulus and Remus (obviously legend) but is there a grain of truth there? He also gives biographies of "all" the kings of Rome (though I suspect that there were more, since Rome is a lot older than legend would have it). Do we know how accurate those descriptions are? If the Republic was founded around the time he claims? Or the other early characters of the Republic which he documents so clearly (some of whom again sound suspiciously like old Mr Livius). Did they really live? How much do we really know for sure about them?
I know that Livy was using earlier sources which are lost to us. So have we any archaeology to confirm or disprove any of his claims?
It's basically impossible to take seriously anything up to and including Ancus Marcius as historical without independent, and much earlier, corroboration (and there isn't any). For the sixth, fifth, and part of the fourth centuries BCE (i.e. the last three kings of Rome and the first century and a bit of the Republic), there's likely to be a certain amount of history mixed in with the mythology, but it's usually very difficult to gauge how much and which bits.
If it's history you want, a good, mainstream piece of reading on the subject is T. J. Cornell's The Beginnings of Rome, which covers all of this period. Chapters 1-8 are on the regal period.
If it's Livy himself you want to study, then the best mindset to adopt is that he starts out 100% fiction, and as he goes on, history gets gradually mixed into the fiction. By the time we get to the Punic Wars (3rd-2nd centuries), we're on reasonably solid ground. Livy himself has this to say about the evidence for everything in books 1 to 5, i.e. prior to the 380s BCE (Livy 6.1):
The history of the Romans from the foundation of the City to its capture, first under kings, then under consuls, dictators, decemvirs, and consular tribunes, the record of foreign wars and domestic dissensions, has been set forth in the five preceding books. The subject matter is enveloped in obscurity; partly from its great antiquity, like remote objects which are hardly discernible through the vastness of the distance; partly owing to the fact that written records, which form the only trustworthy memorials of events, were in those times few and scanty, and even what did exist in the pontifical commentaries and public and private archives nearly all perished in the conflagration of the City. Starting from the second beginnings of the City, which, like a plant cut down to its roots, sprang up in greater beauty and fruitfulness, the details of its history both civil and military will now be exhibited in their proper order, with greater clearness and certainty.
Even after this point there's still an awful lot of imaginative writing: there's zero chance that word-for-word copies of speeches, battle plans, and the like were kept from the 300s and 200s BCE until Livy's time. Asking what was available in Livy's time takes you into the problem of Livy's sources.
Livy rarely names his sources of information, but on the occasions he does, they are for the most part other historians. He's a synthetic writer, not a front-line researcher. It's usually thought that the passage quoted above refers to the pontifical annales maximi ("written records... and even what did exist in the pontifical commentaries"), and that historians of Livy's time assumed (without good reason) that these records had always been kept. His comment about records being destroyed "in the conflagration of the City" (i.e. the Gauls' sack of Rome in "390" BCE -- actually 387 or 386) doesn't mean that he was in the habit of consulting these records himself: in fact even this comment is adapted from another historian's work, either Paulus Clodius or Q. Claudius Quadrigarius (Plutarch Numa 1 = Paulus Clodius fr. 1 FRH). Livy scarcely ever refers to the annales maximi himself, and refers to epigraphic evidence even more rarely.
So his information on early Rome is limited by the quality of the literary sources he had access to. For early Rome, this mostly means Q. Fabius Pictor, who was pretty much the baseline for what the Romans believed about the very early days of their city. Fabius Pictor is lost, so our own information about him is limited by what extant historians (like Livy) choose to attribute to him. As I said Livy names his sources relatively rarely, so it's only occasionally possible to reconstruct any kind of chain of testimony.
There's an introductory discussion of Livy's sources here; for something more in-depth, there's Oakley's treatment in Chaplin and Kraus (eds.) Livy (Oxford, 2009); and there are several relevant chapters in the Blackwell Companion to Livy that's just come out, edited by Bernard Mineo (Oxford, 2014). Edit: If you're feeling more ambitious, you could try opening up one of the major commentaries on Livy and take a look at what the introductions have to say about his sources. And if you want to get a feel for what we know about the kind of historiography Livy had access to, go and open up the Fragments of the Roman Historians (3 vols., ed. Cornell, 2013) and read a dozen or so random pages; all fragments and testimonia come with translations. If you're not used to dealing with lost sources, it should be a real eye-opener.
Edit 2. some minor modifications for posterity.
It's impossible to say, as there isn't really much else in the way of near-contemporary sources with which to check his accuracy. We don't know whether he compiled his books with oral tradition in conjunction with now lost histories, or whether he just made loads of stuff up. There isn't really any way of finding out. Much of his work is a kind of self-justification exercise for Rome, a kind of propaganda tract to show that Rome was an old and established force in Italy, to establish Rome's credentials with the clearly more established cultures of the Mediterranean. One of the problems with Ancient History is the lack of documentary evidence. There are, for example, many Greek Kings from N.India, Pakistan and Afghanistan from about 100 BC to 100 AD of whom we know nothing at all, beyond the fact that coins have turned up with their names on.