The Google Chrome browser has a full screen mode, which you can change regardless of what you have opened. And you can do it only by pressing the F11 key, which is exclusive to the tab in which you are. So you cannot change to a different tab while in this mode.
This is why, if you want to go to a different tab, you have to exit, go to that tab and press F11 again. And if you want to run Chrome in full screen at startup and configure it to always open on a website; you can do it with the use of a «switch». In addition to that you have to make a small change in the browser settings.
See Google Chrome in full screen
In order for you to start Chrome in full screen, you must run it from a shortcut on the desktop. So, if you are a Windows 10 user, you must stop starting the browser from the Start menu.
What you have to do is create a desktop shortcut for Chrome. In Windows 10, you can do it directly from the Start menu. Now, you must open that section and go to the list of applications. Find Chrome and then drag the icon, dropping it on the desktop.
Now, right click on the shortcut and select Properties. Go to the "Shortcut" tab and look for the field called Destination. If you see the end of this field, add a space, followed by the following:
Click Apply and then OK, now you must close the open windows you have. Then, if you click on the shortcut you created earlier, you'll see that Chrome will open in full screen. Thus, and depending on your startup configuration for the browser, a new tab or a specific website will be opened.
Choose which sites you want to open in full screen
If you need to define any web page you want to open with this mode, all you have to do is some changes to Chrome’s settings. Click on the options button in the upper right corner and select "Settings" in that menu.
Scroll down to the "On Startup" section and select the option called "Add a new page." Enter the URL of the website you want to open. This will apply to Chrome as a whole, so if you're not even using the shortcut you created earlier, this website will continue to open.
If this last option is problematic for you, it is a good idea to create a separate profile and configure the home page. You must configure the page you want to open in a different profile. You can also use a switch to run Chrome on that profile you created.
You will not have any problems, since Chrome can be run with many switches easily. So you must place the switch with the full screen and you will have everything as you want.