Any interval cardio at least once a week for your VO2 max. Burpees will help in interval cardio and do help other body parts, but that depends if you are already exercising other parts. In which case you will want them to recover during your cardio so not do burpees. If you are not doing any intensive weights, doing burpees is fine. You will have to rotate it with other exercises for high intensity intervals across 20 mins or so
As we get older, it's important to keep good muscle tone and muscle mass. Burpees will work out mostly your legs (plus cardio) and your shoulders. If I had to do a single exercise (which is not the best thing to do) I would start with planks, which work on your core muscles, and you can gradually improve their difficulty in a variety of ways, by integrating some movements (for example lifting arms and legs, touching your shoulders, etc...).
If you use a computer mouse a lot, draw a lot or do repetitive things with your hands (hammer/drill/rake) then be careful not to over-stress your wrists. Typically if you sit at a desk for long hours you will lose "grip strength" and that is directly related to wrist strength. I like to make a fist to do pushups, but the image below is a good alternative.
How to Prevent Wrist Pain During Push-Ups, Planks, and Burpees