What is causing my circuit breaker to trip instantly and trip the next breaker too? Update - Video Added
Somewhere in the circuit there is likely a low resistance connection between the two legs. The most likely culprit would be a split duplex receptacle that was replaced, and the tab between the upper and lower sections were not removed. Another likely cause would be a staple, nail, screw, etc., that penetrates both cables. 

With both breakers off, check continuity between the ungrounded (hot) conductors of each circuit. 


Turn off both breakers.
Set your multimeter to continuity mode, or the lowest resistance setting.
Touch one probe to the lug of one breaker, and the other probe to the lug of the other breaker.


With the breakers in the off position, you should read infinity (or open).   If you get a reading, it means that these two conductors are connected somewhere.  If the reading in not close to 0, there could be a 240 volt load connected between the legs. If the reading is near 0, you have a low resistance short between the legs.