I have a separate garage, 50' from house. It will have no 220vac, all 120 vac in garage for outlets and lights. Will run 6awg from 100 amp subpanel in basement of house to 60 amp panel in garage. Do I need a grounding rod at the detached garage for the 60 amp panel?
Yes, you will need grounding rods in addition to a wired ground wire in the feeder.
You need 2 grounding rods at least 10' apart.
Do not use a 60A panel in the garage; they have too few spaces in them to be useful. Use a panel with at least 16 spaces; it costs very little more and you'll want the spaces later. The size of the main breaker in the garage panel does not matter, since the 60A breaker at the far end will decide the matter. You need some size of main breaker, because you need a local disconnect switch - and that's the cheapest way to do it.
As far as going 120V instead of 240V, I think that's false economy. You still need to run hot, neutral and ground. The other hot is only one more wire - but doubles your capacity. So crimping yourself down to 120V is false economy.
Speaking of false economy, nothing busts me up more than seeing someone scrimp on a tiny panel or bring out only 120V service "on the logic of saving money", and then turn around and spend a fortune on copper wire because they heard aluminum was ooga-booga. All that's wrong. What you should do is get a big panel, run 240V and use aluminum wire to save money. Aluminum at these large sizes is fine, and always has been. The lugs are aluminum, for Pete's sake.