How do create a routine to print a Delphi TImage together with other Objects on TImage's Parent

You will not be able to make low quality image print as a high quality image.
Think you are going to have to render a high quality image and print that.  You could create a Bitmap in memory and draw your objects to that and then print it or perhaps you could draw directly to the Printer.Canvas.  You would have to scale every thing up and do all of the drawing manually.
Either method is going to require a lot of code as I don't believe there is a simple solution to this without going to third party components.

Tanks for your reply. By high quality print, I mean the print should be at least the quality of what is viewed on the screen, not worse.  Is this still a difficult thing to achieve?  How would one render a high quality image?

The problem is that screen resolution is considerably less than the resolution of printers. When you print the screen capture you are printing using all of the image data that are available. It's just does not look good when scaled up to fill a printer canvans. If you were to print pixel for pixel from the screen to the printer you would find an image with exactly the same quality but it would be quite small.
To draw a higher resolution images you should look at the drawing methods of TCanvas.

This may sound silly, but would there be a way to basically redraw the screen immage onto the TCanvas?  Thus, some code that can recognise lines and coloured blocks, (possibly text will be seen as a series of lines) on the screen immage, and recreate them on the canvas as vector graphics.

While I don't know how you are drawing on the screen to begin with, I can imagine that you could re-use some of your code by simply adding a multiplier in the the drawing methods. Rectangle sizes, line lengths, and font sizes all multiply by 1 on screen and by 6 on printer. The ratio of 1/6 is a guess on my part so you may need to do a little trial and error on that.

As said in the biginning, I am not realy drawing on the screen.  The background is an imported JPG in the TImage picture property.  The small coloured blocks are coloured panels with  the text added in the panel's caption property.  The TImage object is placed on another large TPanel component and the small colured panels also placed on the large TPanel.

What is the resolution (size) of the JPG? Can you print a decent quality versoin of it?

The current bitmap resolution is 800 x 600 and the little panels making up the couloured icons are 40x15.  I think I can thus solve the problem by scaling up the JPS on the external drawing package and also scale the icons up.  Now the second question, lets forget about the print quality for a moment, how does one print the panels (Icons) on the TPanel parent as well as the overlayed TImage containing the JPG picture.

Thanks I think drawing the rectangles is going to be tha answer, I will play around with this and may ask more questions later on.