The two did not meet until 54 BCE, when Antonius, via his connections to Clodius Pulcher, got a spot on Caesar's army staff in Gaul. There he proved himself an able military man, and Caesar both liked him (apparently) and began to groom him for office. Via Caesar's support, he was elected Quaestor in 52 BCE, and then re-assigned to Caesar in that capacity. Antonius led Caesar's cavalry at the famous battle of Alesia in Gaul. After this he seems to have been named legatus by Caesar (an informal position which was appointed, not elected, but carried great responsibility and indirect imperium pro praetore). He remained in that role until the end of the Gallic wars in 50 BCE, often commanding legions in "mopping up operations."

So yes, Caesar and Antonius knew each other for 4 years in Gaul and we have every indication that Caesar liked him and was interested in helping his career. Antonius was boisterous but also a sound and capable commander in the field. They also shared a technical connection through blood, because Antonius' mother, Julia Antonia, was a distant cousin of Caesar.

In 50, Caesar in his guise as Pontifex Maximus got Antonius elected into the prestigious college of Augurs (priests of omens), and then in 49 BCE had him elected as a Tribune of the Plebs. It was during his office as tribune that Antonius vetoed many of the anti-maneuvers of the Senate. They eventually chased him out of the city, after which he re-joined Caesar and participated in the invasion of Italy early in 49.

Antonius was not Caesar's "favorite" lieutenant, however. That distinction was always Labienus, Caesar's very capable right-hand man. Labienus, however, chose to part ways with Caesar during the Civil War and fight for Pompey and the Senate-in-Exile instead. He would command forces against Caesar later at the battle of Pharsalus. Labienus' departure left Antonius among a very small circle of trusted lieutenants who were still around and still pro-Caesar leading up to and during the Civil War. After Caesar's assassination, it was Antonius who stepped up as the logical "heir" of Caesar's forces (a completely arbitrary, non-constitutional notion).

The ancient sources for Antonius are scattered: Cicero has a LOT to say about Antonius, including whole speeches delivered against him. Plutarch has a detailed biography, but of questionable quality. Caesar makes scattered mention of him in bellum civile, but nothing substantial. Same for Suetonius, while describing Caesar's life. Appian also mentions him, but is also a pretty lousy source.