On your phone go to:
Settings = Programmmer = Animations and set them all to zero.

Also disable Wifi, blutooth and Location services if you are not using them at the moment. Delete clean master and cm security as well because they slow down your phone and drain battery fast just like any other task killer app. Delete any battery saver app as well. Go in to your task menu and closed any running app before locking your phone. Hope this helped.

I highly suggest that if your phones boot loader is unlocked and has a custom recovery then you install cm12.1 . If you can do so, you can install a dark theme which makes phone battery last longer and in some cases it can make your phone run smoother/faster. If that's not a option you should use Greenify. I don't think it requires root but if it does, it's not that hard to install and there are probably tutorials for your phone all over youtube. But I don't think it does require root.

they make apps like icleaner from cydia for android, that just clean up unnecessary files, can sometimes do the trick