If I shut down the computer, that opportunity appears as warning. Well, I did it yesterday first time. I have read and understood that it means if I have a site open (in my case wikipedia), I shut down mac and opening it I'll see the wikipedia again. Well, no wikipedia. Mac only opens an one firefox window, which I can do as well myself. Anything wrong? How can I save a page before shutting down computer to continue on that page next day (in case of opening computer again) ?

Reopening windows after a shutdown has never been a 100% proposition -- results vary depending on the macOS version and the applications involved. It usually works for me with Safari (and sometimes Chrome) but maybe not with Firefox..? If you want to be able to reliably get back to certain pages in a browser, bookmark them within the browser (use the browser's options for that, don't depend on that OS feature).

I have never bookmarked anything. I see in up right a page actions pop-up menu gives an option to bookmark this page. I tap there.. ? What else? Does it mean that the bookmarked page opens itself or what the bookmarking means.. ?

Bookmarking simply saves the URL of a particular page, so you can return to it quickly & easily -- the specifics depend on the particular browser you use..