What steps are required to move data from a custom schema in one database to the default schema in new database
You can script out that portion of the database and then run a CREATE DATABASE with the scripted schema and data from your original database.

When you Generate Scripts..., you will have the ability to select specific objects as opposed to the whole database.  You can select the desired schema, and all of the objects that the schema owns.  This will generate a script containing your schema and schema objects.  A bit of a manual operation, but probably the easiest method.

In the generated script, you can alter the schema name to be what the new database's schema should be called, but realize that the object definitions also need to reflect the new schema's name.