Since CyanogenMod is not ready yet for the LG G3, most users will be stuck with the stock OS for some time until CyanogenMod is ready. Until then, a good way to get by is to root one’s phone so you can really control it. There are multiple ways described by #4 in the Sources, but Stump Root has really won the Root competition so far. It does not require a computer, just an apk you can download from the XDA-Developers forum post (Source #1). Some of my experiences with Stump Root v1.2:
The application requires a lot of permissions. Since I was uninstalling the apk after installation, I did not investigate. The application is giving you root access so I assumed it would need all of the permissions.
When I downloaded the apk and allowed Google to check the apk, Google gave me a warning that the application would touch low level pieces of the software, but that is what you want. I suggest not letting Google check out your applications next time.
My phone’s Software Build number KVT49L.D85110m had not been supported yet. The very similar KVT49L.D85110c and T-Mobile LG G3 release firmware were though so I was confident this would work. After pressing Grind, my Build version was not found (expected), but the correct location for app to work with was found in about 10 seconds.
I suggest visiting Source #1 for instructions or #2 or #3 since they will be updated more frequently than this post. I do suggest downloading a root checker before installing SuperSu just to make sure Stump Root worked.