Tap the bookmark icon on the post. Find the list under Bookmarks in the sidebar, or at x.com/i/bookmarks. It is private: other people see a count, not your name. It takes one tap, and it is the right tool for “I might read this later today”. A bookmark is private. Only you can see it (are Twitter bookmarks public?).
It is the wrong tool for keeping the post. The bookmark points at the live post, so a deletion blanks it. Free accounts get one list; folders need a paid plan, covered in bookmark folders. There is no export button. The longer version is the X bookmarks guide.