Johnny first hears the poem reveals that Johnny has more depth than we have seen previously. The reader already knows that Ponyboy is academic and sensitive, with an intuitive understanding about the people around him. Therefore, it is not surprising that he marvels at the beauty of nature as he watches the sunrise. We have seen this side of Ponyboy before, when he and Cherry watch the moon and he thinks about how universal it is.

However, we get a new side of Johnny in the scene, as he is equally impressed by sunrise. He says,

"Golly … that sure was pretty … The mist was what was pretty … All gold and silver."

He tells Ponyboy,

"I never noticed colors and clouds and stuff until you kept reminding me about them. It seems like they were never there before."

At the sight of the sunrise, Ponyboy recalls the Robert Frost poem, which he recites to Johnny. Johnny is impressed. Later, in the hospital, just before he dies, he tells Ponyboy to "stay gold," which suggests that he understands the significance of the line in the poem, "Nothing gold can stay." His reference to this line refers to the brief and ephemeral stage of young adulthood when a person has their entire life ahead of them.

He means for Ponyboy to hold on to his innocence and youth as long as he can. The source of quote is a poem by Robert Frost, "Nothing Gold Can Stay."

I assume you are referring to The Outsiders.  In The Outsiders, there is reference to the following poem:

Nature's first green is gold,
Her hardest hue to hold.
Her early leafs a flower;
But only so an hour.
Then leaf subsides to leaf.
So Eden sank to grief,
So dawn goes down to day.
Nothing gold can stay.

This poem is called "Nothing Gold Can Stay" by Robert Frost.  The meaning of the poem is that everything starts out young and innocent, but it cannot stay.  Good is "the hardest hue to hold" because there are so many influences just from living life that can corrupt us.  When Johnny tells Ponyboy to "stay gold" he means that he wants him to stay good, and not be corrupted by the negative forces in the world.

The Outsiders is a coming of age story.  Although Ponyboy is a good person, he gets caught up in gang life and ends up going on the run after a boy dies.  Throughout the story, Ponyboy keeps his good heart and does manage to stay gold.

Johnny told Ponyboy to stay gold to remind him that he did not need to stay in the gang life.

Although Pony describes the greasers as protecting each other like family and assures the reader that the kids in his town divide their affiliation by socioeconomic status, the greasers are still a gang.  The often get into trouble, with the law or with the other gang, the Socs.

Pony tells us that greasers are poorer than Socs, and also “wilder.”

Not like the Socs, who jump greasers and wreck houses and throw beer blasts for kicks … Greasers are almost like hoods; we steal things and drive old souped-up cars and hold up gas stations and have a gang fight once in a while. (Ch. 1)

Although he says he does not do those things, they are part of the life of a greaser.  Pony is expected to participate in these activities eventually, especially fighting.  He says that greasers can’t walk alone for fear they will be jumped by Socs.  

Pony is different from the other Socs.  He does well in school and likes to read.  He is just a deep thinker overall.  Johnny is aware of this.  When he and Pony spend time on the run after Johnny accidentally kills a Soc, Johnny brings Pony a book and the two spend their time discussing the novel and poetry.

While on the run, in addition to reading Gone with the Wind, they discuss the Robert Frost poem “Nothing Gold Can Stay.”  The poem is about young things in nature not lasting, but it also has a metaphorical quality for Pony.  It means that as you get older, you get more corrupt.  In the hospital, Johnny tells Pony to stay gold.

"We told him about beatin' the Socs and... I don't know, he just died." He told me to stay gold, I remembered.  What was he talking about? (Ch. 10)

Johnny means that Pony has the potential to get out of the gang life. He can get an education, leave town, and become a responsible and contributing member of society.  With Johnny’s last words, he reminds Pony that the gang life is dangerous and while not all greasers have a choice, he does.

What Johnny is saying to Pony is that he does not want him to change.  He does not want Pony to become hardened and bitter.  He thinks that Pony has the potential to become something and he does not want him to end up like, for example, Dallas Winston.

In the poem, Frost says that things that are beautiful (like gold) cannot stay that way.  He says that the first leaves that come out are the most beautiful, but they too cannot stay that way.  Johnny is trying to encourage Pony to resist that trend and to stay beautiful (inwardly) and good.

At the end of the story, Ponyboy discovers Johnny's letter in the novel Gone with the Wind, which inspires him to write the novel The Outsiders in order to share his story with other kids struggling to overcome similar obstacles in their lives. In Johnny's letter, he interprets the Robert Frost poem by explaining to Pony that staying gold is equivalent to remaining youthful, innocent, and hopeful in life. Johnny once again encourages Ponyboy to stay gold and to share the same message with Dally, who desperately needs to see the positives in life. After reading Johnny's poem, Pony feels inspired to help the thousands of adolescents throughout the country dealing with similar issues and immediately calls his English teacher to ask him about the required length of his theme assignment. Pony then thinks about Bob Sheldon, Dally, and Johnny before beginning to write the story that he just narrated for his English assignment. Pony's first words of his theme assignment are the first words of the novel The Outsiders. Overall, Pony shares Johnny's message with his English teacher and countless adolescents who end up reading the novel.

Yes, Ponyboy does plan on relating Johnny's message about 'staying gold' to other people, as revealed in the final chapter of the novel:

"I could picture hundreds and hundreds of boys living on the wrong sides of cities [...] Someone should tell their side of the story, and maybe people would understand then and wouldn't be so quick to judge a boy by the amount of hair oil he wore" (179).

Ponyboy contemplates the importance of telling the other boys' stories after finishing Johnny's letter.  He wishes he could tell Dally, knowing and hating the fact that it was too late.  After considering all the other boys who might need to hear a similar message, Ponyboy picks up his phone and called his English teacher, Mr. Syme, wondering about the required length of his English theme essay.  The reader can infer that Ponyboy plans on telling his and Johnny's story for his English paper that is due.