If you just want to get going with a code-first database, you don't even need to define a connection string -- just write your objects.

Named instances: they are private and are owned by a single application that is responsible for creating and managing the instance. Now, connection string for public localdb should look like ...

Untangle SQL Server connection strings Today’s README file explains how to create connection strings to SQL Server databases using Visual Studio’s built-in tools.

With Visual Studio 2022, I had no issue connecting to (localdb)\MSSQLLocalDB (SQL Server 15), so I discocovered in Visual Studio the connection string of the localdb instance.