How do I use the Enum value from a class in another part of code?

Use

core.gameMode = Core::INIT;

The individual values of an enumeration are scoped not within that enumeration but at the same level as the enumeration itself. This is something that most other languages (including C#) do differently, and C++0x will allow both variants so that there,

core.gameMode = Core::GAME_MODE::INIT;

will also be legal.

In addition, the strongly typed enums that will be added in C++0x (enum class) will put the enum values only within the scope of the enum (i.e. as in C#); this solves the problem you noted in your comment that for “normal” enums, the identifiers for enum values need to be unique across all enums defined in the same scope.