How would a person use the Lua scripting language to do things such as this:
Opening an application such as IE and controlling it, accessing the C drive and the curr
Generally speaking, Lua doesn't have this kind of stuff built-in. Lua is a minimalist programming language, great for embedding. The core language only uses facilities available in C89. If you want bells and whistles, you need to add Lua libraries.
You can find libraries by searching the wiki, at luarocks, luadist, or luaforge
For "controlling IE" you should take a look at LuaCom
For accessing the current directory, try LuaFileSystem