Harvey, the first thing I ever did involving electronics was program a micro controller, and I knew nothing at all when I started. (ahh yes, my first blinking LED I remember it well) PIC or AVR(my personal choice) doesn't really matter, pick a cheap general purpose device 8-40 pins in a DIP package to start cold, you just need a breadboard and some jumper wire, and whatever else you can scrounge from other defunct electrical equipment. With modern MCU's what you can do with them is truly limitless, because once you get used to them and their limitations you'll realize you need support circuitry to drive real world loads and you'll have to learn how those work. I wouldn't discount discrete logic for the life of me, but MCU's are really trival to work with nowadays, and getting up to speed on current trends is a good way to get back into electronics after 20 years I'd think.
So my suggestion would be to carefully define what you're getting back into electronics for. Fun, learning, idle time? You could pick right up wherever you left off with your knowledge of electronics but you might be missing starting with what's currently going on, advances over the last 20 years are truly staggering. Just don't be afraid to try something new.
People here tend to tackle a problem head on and full bore before understanding the intent of the original poster. So it's really up to you. You can find on the net equally well how to make a stop watch clock such as you describe using both discrete logic with a basic watch crystal to get you 1/100ths of a second, or a 20mhz+ MCU to get you 50 nanosecond accuracy.