The Samsung Galaxy Note 3 is a superb device and one of the best to come from the phone maker. However the look of it is nothing to shout home about, but you can put it into a cover. Along with giving it a new look, a cover offers protection. So here are some cases that you may wish to check out.
If you want a skin type case then you could take a look at the Cruzerlite AndroidField Clone Army case, the Bugroid Circuit case, or the A2 TPU case. The benefits to these cases are that they are thin, light and flexible.
For hard cases you may wish to check out the Mobi Hard Shell Case, Incipio Feather SHINE Case, Seidio SURFACE Case (with kickstand), Case-Mate Carbon Case or the Cruzerlite DNA Case.
If you prefer a holster type case then take a look at the DILEX Holster, SURFACE Holster, and Seidio Spring Clip Holster.