What is the resulting set for {0,1}*\{0}*?
You are correct that the words not in $0^*$ are exactly those words which contain $1$ somewhere. You are also correct that you can represent this complement using the regular expression $(0+1)^*1(0+1)^*$. There are other regular expressions for the same language, for example $0^*1(0+1)^*$ (which is a "deterministic" regular expression).