A 20+-year-old steel bike (I have two of them) is apt to have suffered some rust damage along the way. If you still have the frame, look inside to see if the stay rusted almost through before it broke. I have a mid-'80s Trek I converted to singlespeed a few years ago, and I really like it, but I'm increasingly reluctant to ride it because, while the outside looks good, I don't know what's inside the tubes.
On the new bike, miss what potholes you can, "float" over bad spots by standing and absorbing shock with your legs and arms, and run the biggest tires you can at fairly low pressure. The bike I ride most often in the city has 700x37 tires, and most of the time I keep them at 70psi.