Cmd-Ctrl-Space (⌘⌃␣) for the character/emoji picker. Also, under Keyboard in preferences, check the "Show keyboard and emoji viewers in menu bar" option to get it as a menu-bar icon. This also includes the keyboard viewer, which is handy to figure certain Shift-Opt-Whaterver combos for seldom-used characters. EDIT: Updated with the actual default command.
I think this may not be the default; it doesn't work for me.
While speaking of the character picker, does anyone know a way to get less broken search in the picker? For example, I frequently find myself looking for math italic characters in Unicode. Despite the description being, say, "MATHEMATICAL ITALIC SMALL A", it doesn't come up when I search for 'small'. In this case searching for 'mathematical' or 'italic' works, but there are other cases where I just have to guess search terms randomly (or go look up Unicode tables elsewhere).