UPDATE:
My advice below tells you how to script a DB using SQL Server Management Studio, but the default settings in SSMS miss out all sorts of crucial parts of a database (like indexes and triggers!) for some reason. So, I created my own program to properly script a database including just about every type of DB object you may have added. I recommend using this instead. It's called SQL Server Scripter and it can be found here:
https://bitbucket.org/jez9999/sqlserverscripter
I'm surprised no-one has mentioned this, because it's really useful: you can dump out a database (its schema and data) to a script, using SQL Server Management Studio.
Right-click the database, choose "Tasks | Generate Scripts...", and then select to script specific database objects. Select the ones you want to copy over to the new DB (you probably want to select at least the Tables and Schemas). Then, for the "Set Scripting Options" screen, click "Advanced", scroll down to "Types of data to script" and select "Schema and data". Click OK, and finish generating the script. You'll see that this has now generated a long script for you that creates the database's tables and inserts the data into them! You can then create a new database, and change the USE [DbName]
statement at the top of the script to reflect the name of the new database you want to copy the old one to. Run the script and the old database's schema and data will be copied to the new one!
This allows you to do the whole thing from within SQL Server Management studio, and there's no need to touch the file system.