One Circuit Tripping Another Circuit
You either have a short or you are back feeding from another circuit. I'm not sure, but let's start at the Panel. Since you know which breakers are tripping,  turn off one circuit breaker and check the load side of the that breaker and see if we are reading a voltage. Try it again in reverse with the other breaker. If you are reading voltage with the breaker in off position then you know you have a phase to phase sort somewhere in the wiring. You simply have to go out and find where they are connected together. By the way use a voltmeter. 

I would be looking for a white wire that is being used as a part of a switch leg that is somehow feeding back into a neutral.

If you have no voltage as discussed, then disconnect all equipment and check to see if it is the wiring or a piece of equipment. If it's the equipment, get it replaced. If its the wiring we are back to tracing the wiring out.

What ever you are doing just make sure the power is of before working on it. I would only have the power on when I was looking for voltage. 

I am assuming you installed the 5 circuits yourself and are familiar with what was done. If you had an electrician do the work, just call him and have him come back and fix his problem.

Good luck and stay safe.