I've tried pasting "圍棋 (wéiqí)" into my editor, but after typesetting I see only "?? (wiq)" in the PDF. What do I need to change to fix these characters?
You need to use an TeX engine (XeTeX or LuaTeX) that handles Unicode input correctly and a font that contains the characters. You also need to make sure that your file is saved as UTF-8.
See the following questions for help on these issues: