How can I figure out why my breakers are tripping?
First, breakers are designed for avoiding draw over the rated current - old fuses did this by melting when the line in the fuse got too hot and melted.  They are not designed as switches for operating a circuit, so your statement "It's not a big deal to flip the breaker" is only in reference to the amount of inconvenience you suffer.  The breakers should not be flipping constantly.

The reason most breakers flip is when you run a motor that pulls more current than the breaker is rated for, so the simple fix if things were wired correctly would be to use a lower amperage motor or a higher amperage breaker.  The risk with the higher amperage breaker is that the wiring in the wall is of insufficient gauge and will generate enough heat due to resistance to melt the insulation and cause a short.

The fact that the motor is apparently on more than one branch circuit is the primary cause for concern.  This is a wiring issue, and can be solved by isolating the motor on its own new branch circuit.  That will not solve the underlying issue of having (probably) multiple instances of poor wiring.

As far as diagnosing the specific problem goes I would recommend hiring an electrician to trace the branch circuits involved (you could do the same with a continuity tester and a few long leads) and write up a proposal to fix the wiring issues.