I have figured out how to send the input to the console.  I used what Jon Skeet said.  I am not 100% sure this is the correct way to implement this.
If there are any comments on to make this better I would love to here.  I did this just to see if I could figure it out.
Here is the program I stared that waited for input form the user
class Program
{
    static void Main(string[] args)
    {
        // This is needed to wait for the other process to wire up.
        System.Threading.Thread.Sleep(2000);
        Console.WriteLine("Enter Pharse: ");
        string pharse = Console.ReadLine();
        Console.WriteLine("The password is '{0}'", pharse);
        Console.WriteLine("Press any key to exit. . .");
        string lastLine = Console.ReadLine();
        Console.WriteLine("Last Line is: '{0}'", lastLine);
    }
}
This is the console app writing to the other one
class Program
{
    static void Main(string[] args)
    {
        // Find the path of the Console to start
        string readFilePath = System.IO.Path.GetFullPath(@"..\..\..\ReadingConsole\bin\Debug\ReadingConsole.exe");
        ProcessStartInfo startInfo = new ProcessStartInfo(readFilePath);
        startInfo.RedirectStandardOutput = true;
        startInfo.RedirectStandardInput = true;
        startInfo.WindowStyle = ProcessWindowStyle.Hidden;
        startInfo.CreateNoWindow = true;
        startInfo.UseShellExecute = false;
        Process readProcess = new Process();
        readProcess.StartInfo = startInfo;
        // This is the key to send data to the server that I found
        readProcess.OutputDataReceived += new DataReceivedEventHandler(readProcess_OutputDataReceived);
        // Start the process
        readProcess.Start();
        readProcess.BeginOutputReadLine();
        // Wait for other process to spin up
        System.Threading.Thread.Sleep(5000);
        // Send Hello World
        readProcess.StandardInput.WriteLine("Hello World");
        readProcess.StandardInput.WriteLine("Exit");
        readProcess.WaitForExit();
    }
    static void readProcess_OutputDataReceived(object sender, DataReceivedEventArgs e)
    {
        // Write what was sent in the event
        Console.WriteLine("Data Recieved at {1}: {0}", e.Data, DateTime.UtcNow.Ticks);
    }
}