I've traveled with my phantom 3 several times. I use the DJI branded Monfrotto backpack (sorry not an ad) and I like it for convenience (backpack style) and accessibility. Security - I'm TSA Pre-check, so I typically get to go through short lines, but be prepared to be stopped. I've been through security several times and am about 50% on getting stopped by them wanted to see more. It only adds about 5-10 minutes to the line, but dont cut your flights close and they've never told me no. I always carry the drone, extra props, 2-3 standard Phantom 3 batteries, 2 chargers, and all the smaller accessories. I dont take knives/screwdrivers as TSA generally doesnt allow those.
As for on the plane, I'm generally overprecautious, but I take the batteries out of the bag and put them in the seat pocket. I also inform the flight attendant(s) who generally say just to keep it accessible (easier to handle if they are out in the open as opposed to stowed in the overhead bin). I do keep a fireproof bag in the seat pocket as well, but I have never used it/put the batteries in it. No issues so far and have made about 4 trips with the drone.
I do know you are NEVER to put the LIPOs in checked baggage, so make sure if they gate check your stuff, you pull them out. Again, back to the backpack, it is small enough to fit under the seat for any aircraft I fly so I dont have to worry about that getting gate checked. I've never done overseas, but will be heading to Hawaii later this year, so will see how that goes.
Best of luck!