How do I resize an image without losing its quality?

{
    OpenFileDialog opendialog;
    SaveFileDialog savedialog;
    Image target_image;
    int width;
    int height;

    public Form1()
    {
        InitializeComponent();
        opendialog = new OpenFileDialog();

        opendialog.RestoreDirectory = true;
        opendialog.InitialDirectory = "C:\\";
        opendialog.FilterIndex = 1;
        opendialog.Filter = "jpg Files (*.jpg)|*.jpg|gif Files (*.gif)|*.gif|png Files (*.png)|*.png |bmp Files (*.bmp)|*.bmp";

        savedialog = new SaveFileDialog();

        savedialog.RestoreDirectory = true;
        savedialog.InitialDirectory = "C:\\";
        savedialog.FilterIndex = 1;
        savedialog.Filter = "jpg Files (*.jpg)|*.jpg|gif Files (*.gif)|*.gif|png Files (*.png)|*.png |bmp Files (*.bmp)|*.bmp";
    }

    private void btnOpen_Click(object sender, EventArgs e)
    {
        try
        {
            if (opendialog.ShowDialog() == DialogResult.OK)
            {
                target_image = Image.FromFile(opendialog.FileName);
                width = target_image.Width;
                height = target_image.Height;
                }


            target_image = Resize(width, height);


        }
        catch (Exception ex)
        {
            MessageBox.Show("Error Occured, " + ex.Message);
        }
    }

    private new Bitmap Resize(int target_width, int target_height)
    {
        target_width = 300;
        target_height = 500;

        Rectangle rectangle = new Rectangle(0, 0, target_width, target_height);
        Bitmap destImage = new Bitmap(target_width, target_height);
        destImage.SetResolution(target_image.HorizontalResolution, target_image.VerticalResolution);
        using (var g = Graphics.FromImage(destImage))
        {
            g.CompositingMode = CompositingMode.SourceCopy;
            g.CompositingQuality = CompositingQuality.HighQuality;
            g.InterpolationMode = InterpolationMode.HighQualityBicubic;
            g.SmoothingMode = SmoothingMode.HighQuality;
            g.PixelOffsetMode = PixelOffsetMode.HighQuality;

            using (var wrapmode = new ImageAttributes())
            {
                wrapmode.SetWrapMode(WrapMode.TileFlipXY);
                g.DrawImage(target_image, rectangle, 0, 0, target_image.Width, target_image.Height, GraphicsUnit.Pixel);
            }
        }
        return destImage;
    }

    private void btnSave_Click(object sender, EventArgs e)
    {
        if (savedialog.ShowDialog() == DialogResult.OK)
        {
            target_image.Save(savedialog.FileName);
        }
    }
}