I have a list which have several content types attached to it. When you click on new item I want a dropdown box to appear where you can change the content type and it then changes the fields that can be displayed. I know something like this is possible as it is used in the Asset Library.
That option only shows up once an item has already been added to a list or library. In your custom list with multiple content types, add an item and then edit it. When you edit it you will see the option to change its content type.
In the Asset Library, you are first uploading the file and then editing its properties. Thus the ability to change the content type shows up.