Flagstaff and GCNP......might choose to go there on Amtrak,Can I get around without
 

rental car?

You will be fine at the Grand Canyon. You can actually take a shuttle from Phoenix to GCNP. And a car is a hindrance since you have to park and take the shuttle everywhere anyway. Not sure about Flagstaff. But its a college town so it should have a bus system??

Groome Transportation runs 3 shuttle trips per day from Flagstaff to Maswik Village in Grand Canyon NP. They do have a stop at the Amtrak station in FLG. Round trip is in the $65-$70 range. A rental car where you can set your own schedule may be a better deal.
Are you coming from the east (Albuquerque) or from the west (LA)?

While it's possible as described above to get to GC & Flagstaff without a car, and if you really want to do it all by train, there is a train (link below) from Williams (served by Amtrak) to GC, I think you'd need a relaxed schedule to accommodate the train & shuttle schedules. Otherwise, the difficulty as I see it is adequate means of getting around while in Flagstaff. The Amtrak station is downtown, which is compact and would be enjoyable without a car, but there are some interesting things to see around Flag that would only be practical with a car. The information in your post is limited so it's difficult to say if having a car would be better or if trains & shuttle would be enough. A compromise might be to rent a car for a day or 2 while in Flagstaff.

https://www.thetrain.com/the-train/schedule-route/

I ran across this link looking for the GC train website, some with both GC & Flagstaff:
https://www.amtrakvacations.com/dest...n/williams-az/