How do I get full screen display when I open Chrome?

I think what you want is to go to System Settings > Desktop & Dock > Menu Bar > Automatically Hide and show menu bar and set to Always

What do you mean it doesn't open in full screen exactly?

For full screen you can click on the green bubble on the top left corner of the Chrome window.

If you want to enlarge the window, move the mouse cursor to the edge of the screen, until it turns into a double headed arrow, then just click on the edge or corner of the Window, and drag it away to make the window larger.