The apps that has been lagging for me yet are:
1- Facebook Messenger: the lag is in the pop up animation when the app notifies you on the screen and sometimes it doesn't lag.
2- Google Play: its always lagging but this is due to the caching process while you load the list of items it takes a while to show up the related icons and stuff so this is the cause of lagging.
3- While you are working with downloads and installations its always lagging in any situation. Transitions, using other apps and so on.
4- When you browse the internet sometimes it lags.
I think the lag is not due to the hardware quality and those stuff that people are considering. I think its a matter of 1 or a combination of the following:
1- The CPU.
2- RAM Management
and this is because if you restarted the device you will be able to use it lag free for a while.
My best practice with the battery is that I use it always down to 2%, shutdown the phone totally and leave it in the charger, never overcharge it I don't trust the way Samsung handles that and then use it all day long with everything working like a charm.
Some solutions are about the developer options and turning on animations but I think its just to hide the lag not to remove it so I don't do it. I always clean up the cache, Removing all unwanted applications, games and useless background services and using the task manager to end unwanted running applications from time to time. If you do so I guarantee you the best user experience with you Note 3
I will be reporting all issues for you and hope that the next 4.4.2 update handles all this sh*t