Go your own way is perfect. You can find thousands of examples of how people use it in English on Google Books. There is even a well-known song by Fleetwood Mac called "Go Your Own Way".
Below is some explanation of why "Go your own way" is correct and the others are wrong or not as good.
"Go your own way"
In that sentence, "your own way" serves as an adverb modifying "go". The same kind of phrase used as an adverb appears in sentences like:
Do it your way.
Google Books has thousands of examples of that, too. There is even a famous song by Frank Sinatra titled "I Did It My Way". (It's probably his most famous song.)
"Go to _____"
When you say "Go to _____," normally after "to" we expect a noun that names the destination. Here are some examples:
Go to your room. [This is what parents say to their children when they are misbehaving. It means that the child should go to his or her bedroom, away from the family activity that he or she was disturbing.]
If you commit a crime, you might go to jail.
"Go in your own way"
Go in your own way is correct, but not as strong. The preposition in can introduce almost anything to serve as an adverbial phrase, whereas "your way", "my way", "her way", "that way", etc. are somewhat special noun phrases that can serve as an adverb without a preposition.
Go your own way echoes familiar songs and phrases that excite emotions and evoke deep elements of the individualistic culture of America. Go in your own way does not.
Some complexity that you should understand
You should know that "Go your own way" is not necessarily positive. It can emphasize being alone and therefore unaided by other people. In the Fleetwood Mac song, it refers to breaking up a romantic relationship. The full sentence in that song is "You can go your own way", which in that context means "I don't want to be your boyfriend/girlfriend anymore." Even this isn't completely negative, though. "Go your own way" rather than, say, "Go to Hell", suggests that each person has a separate path in life, which each should follow, and continuing to stay together would prevent that.
Trying to learn the full meaning of these phrases from simple, abstract definitions will only mislead you. To really understand "your way" in the context of the father encouraging the son, you should listen to the whole Frank Sinatra song and read the words. The meaning is complex but is well expressed in the song: when you go your own way, you struggle more, you're defeated more, and you need more courage than when you follow the crowd, but the journey is more fulfilling—or at least this belief is held deep in the American worldview, even by the many Americans who follow the crowd.