after installing SQL Server Express:

1. install SQL Server Management Studio Express

2. create your database in SSMS Express, give it a logical name (e.g. MyAppDatabase)

3. connect to it using its logical database name (given when you create it on the server) - and don't mess around with physical database files and user instances. In that case, your connection string would be something like:

   Data Source=.\\SQLEXPRESS;Database=MyAppDatabase;Integrated Security=True
   

   and everything else is exactly the same as before..