Animations in my new Moto G 3rd gen are very laggy,even compared to my old Moto E2 4G.Also,Free RAM is just 750 odd MB..

Don't be concerned about the amount of free RAM you have.  Android does a very good job of memory management and will allocate RAM as required.

Please give us more information about the animations.  Which apps?  What are you doing when it happens?  Any details would be helpful.

Does the phone lag if you run it in safe mode?

Yes,it does..of course not much compared to when not in safe mode..I also did a factory reset..lag still persists..

When you did the factory data reset, did you have automatic restore turned on?

I did factory reset about three days ago..and I had automatic restore on...it was alright the first two days after reset..but today lag started again in animations and transitions..

If you had automatic restore on, then if there's an app causing the problem, you might have reinstalled the problem.

Can it be the Greenify sir..??

If you think it might be Greenify, uninstall it and see what happens.

Whenever somebody says that an FDR makes the phone run okay for a while and then it starts misbehaving, I always think the first place to look is a third party app.  If the problem was with the operating system, or with hardware, the phone wouldn't run properly no matter what you did.