Antonio is a Veronese nobleman and the father of Proteus. In the play, he decides to send his son to travel abroad (to Milan) so that Proteus will have a well-rounded education:
I have consider'd well his loss of time
And how he cannot be a perfect man,
Not being tried and tutor'd in the world (1.3.3)
Antonio gives voice to a pretty common Renaissance belief – that a man's education is incomplete (thereby making the man incomplete or "[im]perfect") if he doesn't venture out to see the world.
This, sadly, is the last we see of Antonio, which is too bad because we'd really like to know what Antonio would have to say about whether or not his son learns anything while he's off being "tutor'd in the world."