I have a SQL connection string being stored in the web.config.
How can I pull items like server name, username, password... inside of C# ASP.Net code?
You can use the SqlConnectionStringBuilder for that:
var connectionString = GetConnectionString(); //e.g. use code shown in other answer
var builder = new SqlConnectionStringBuilder(connectionString);
var user = builder.UserID;
There's some sample code on this MSDN page.
ConfigurationManager.ConnectionStrings("name of connection string in web.config")
Here is a link to the msdn documentation.
Enjoy!