Not to be a jerk, but maybe UTFG? A Google search for "starting cycling at 60" gives me 98,000,000 results in 0.47 seconds.

IMO, this forum would be much more useful to you if you read some of that info first, and then came to the forum with more specific questions, like "I'm starting as a cyclist at 60+, I've been advised to do x, what are your opinions about this advice?", or similar. With the broad question you asked, you're going to get advice that is all over the place.
- Read up a bit.
- Consult your Dr. about starting an exercise program if you have any health problems or, at your age, any family history of health problems.
- Pick or cobble together a "program" that sounds good/fun to you.
- Ask questions about it.
- Try it out.
- Use feedback from your rides and the forum to adjust your "program" to fit you.
- If you find you don't like cycling, try something else. You'll be more likely to stick with something you enjoy. Running? Hiking? Rowing? XC skiing? Or multiple activities to keep things interesting (also a great way to support the economy, you'll buy lots of gear!!).