There's not a lot left to know. Grab yourself Plutarch's or Suetonius' biography; both are available in decent English translations on the internet. His father died when he was 15; married young to the daughter of Sulla's enemy Cinna; when he refused to divorce her he got himself proscribed. We know about his run-in with pirates and his rumored affair with the king of Bithynia from several sources; those stories are in Plutarch and/or Suetonius among other places.

Goldsworthy's account of Caesar's early life relies on the general approach Romans took to aristocratic child rearing. Probably had a wetnurse. Probably educated at home. Probably taught Greek as well as Latin. His tutor Gnipho had studied in Alexandria.

A quick aside, Goldsworthy has this weirdly modern sentence:

Caesar was raised to think of himself as special. In itself this was nothing unusual...

I wonder whether he's projecting modern concerns into the past here.

When he was old enough he would have trained on the Campus Martius, attended law courts and such. After comments like this, Goldsworthy goes on to that business with Sulla.

Meier's biography is a bit older, and has been criticized for plenty of errors, but it's a better read even in translation. I'm not sure it's much more help though. He starts his chapter on Caesar's upbringing with

Almost nothing is recorded of Caesar's childhood and youth.

and he also relies on a kind of general understanding of Aristocratic Roman upbringing. He mentions rhetoric, Greek, the Twelve Tables, riding, swimming, that sort of thing. Unlike Goldsworthy, he stresses the influence of Caesar's father as a model for imitation. And after that, he turns to Caesar's military career.

So, that's about all we can say before Caesar becomes politically active and Sulla chases him out of Italy.