How do public shares work? - Company going public?

I am getting confused on exactly what it means for a company to be public. If people can help me understand this concept that would be great! Thanks in advance.
So say me and a friend each own 50% of a private company. Now that we want to take the company public first and later have a secondary offering:

1. My friend wants out by selling his shares in an IPO. So 50% of the company is going public. Lets say 50% is 500 shares and each share is sold for $10 dollars. Would that mean the market cap is $5,000? And since only 50% of the company is technically public, doesn't the market cap NOT represent the equity value of the whole company? How does that work? Is the market value of equity of public company mean 100% of the ownership of a company? Where are my shares calculated?

2.. I now want out of the company as well, does that mean I do a second offering of "old" shares and sell my stake? How would that work and how many shares would there be afterwards?

Hope people can help me out. Thanks!