Welcome.
I do not think 60 is old at all. I bought my first bike in the middle of June-first time ever on a bike at the age of 37. It was a "mode" bike, by Giant called Giant Suede W. It was more bike than I wanted. Then 3 weeks later I wanted more power-who could have guessed? Not me. 37 years being terrified of bikes...So I had this nice bike, and wanted a new one. I ended up with a Trek 1000 and I love it. I spent $650 on it. Trek makes some lovely comfort and hybreds for less. Do I have a point? Think about what you may want to do. It hurt financially to buy another bike just 3 weeks later. Fortunately the bike store where I bought my Trek was willing to give me store credit for my Giant (bought at another LBS).
Best advice, think about your needs/future goals, and go try out as many bikes as you have time for. When a bike fits, you will know it. It will just feel as though it belongs under you.
Above all, have fun!
Jennifer
“Live as if you were to die tomorrow. Learn as if you were to live forever.”
-Mahatma Gandhi
"We are what we repeatedly do. Excellence, therefore, is not an act but a habit."
-Aristotle