Not a problem at all to travel with your drone. TSA is perfectly fine with it, with a few caveats... But nothing major.
Has to be carry-on. You can only carry two batteries with you, depending on size, and no they don't have to be discharged.
I personally like to take the props off and put them in my regular luggage that goes underneath the plane so there's no issue with the blades being considered harmful, and to give them an extra bit of confidence I won't power it up at any point during my travel.
Spirit Airlines on Drones:
Can I carry a DRONE onboard?
TSA Rules on Drones:
Drones
TSA Rules on Lipo Batteries:
Lithium batteries with 100 watt hours or less in a device