Actually there are quite a few ways to create a database connection, but I would say one of the easiest ways is to utilize the methods and classes found in System.Data.SQLClient. A basic connection would look something like the following:
using System.Data.SQLClient;
namespace YourNamespace
{
public class DatabaseConnect
{
public DataType getData()
{
DataType dataObj = new DataType();
SqlConnection testConn = new SqlConnection("connection string here");
SqlCommand testCommand = new SqlCommand("select * from dataTable", testConn);
testConn.Open()
using (SqlDataReader reader = testCommand.ExecuteReader())
{
while (reader.Read())
{
//Get data from reader and set into DataType object
}
}
return dataObj;
}
}
}
Keep in mind, this is a very, very simple version of a connection for reading data, but it should give you an idea of what you need to do. Make sure to use a "using" or "try/catch" statement to ensure that the connection is closed and resources are freed after each use (whether it successfully gets data or not).
As for your other question about what equipment you may require. In the beginning I would suggest just creating the database on your local machine and running tests from there. Once you are confident with trading data back and forth, feel free to move the database to another box or an online server. Any internet connection type should suffice, though I can't vouch for dial-up, haven't used it in years.
One final note, if you do happen to decide to move to an online server system, make sure that the service you use allows for outside connections. Certain services use shared server systems, and force users to use their internal database interfaces to manage and write to the database.
--- EDIT ---
As for the server system itself, build up a separate box on your local network that you can see, and load up the database software of your choice. Since you are using C#, it would probably be easiest to go with Microsoft SQL Server 2005 / 2008. The installation is rather straightforward, and it will prompt you to automatically create your first database while installing.
After installation it will be up to you to add in the tables, stored procedures, custom functions, etc... Once your base structure is created, go ahead and use the above code to make some simple connections. Since you are familiar with the above practices then I'm sure you know that all you really need to do is target the server machine and database in the connection string to be on your way.