How do I create a new user in SQL Server 2012 that I can use in a connection string?

Before you add the user to the db, add the user to the server -- under the server, right click the security folder, and try it that way. Then you can grant permissions to the appropriate databases to that user

Here's SQL way of doing this:

CREATE LOGIN NewUser   
    WITH PASSWORD = '340$Uuxwp7Mcxo7Khy';  
GO  

  CREATE USER NewUser FOR LOGIN NewUser;  
GO  

EDIT: As per comment by @bschipp

Another example that does not expire password would be following

CREATE LOGIN NewUser   
    WITH PASSWORD = 'Uuxwp7Mcxo7Khy$#' CHECK_EXPIRATION = OFF, CHECK_POLICY = OFF;
GO
    CREATE USER NewUser FOR LOGIN NewUser;  
GO  

more on options you can read here.