Am I better off buying a new bike, or upgrading my old bike?
As @Daniel said - I would suggest looking for a newer second hand bike closer to what you want. the cost of components, along with compatibility issues, will like make the exercise more expensive with a bike that has a few niggly problems that just won't go away.  However, it you are particularly attached to the frame, it might be worth while. Consider getting a donor bike, as whole bikes are usually significantly cheaper than the sum oft there parts.

If you go down the seconds hand or new path, Jamis (an I am sure others) make mid and high quality steel bikes (Look for the Dragon range in MTB) for those of us that appreciate the advantages of steel. At the highest end, the weight penalty is so small it can be ignored. 

Edit :  List of steel road bikes