What to look at first when when circuit breaker that fridge is on is tripping?
The simplest, cheapest, and fastest test would be to swap the wire into the fridge breaker with another one of the same rating.  (Turn off both breakers first.)  This would test if the breaker has "de-rated" itself.  That sometimes happens as they age.  If the swapped arrangement doesn't trip, replace the (original) refrigerator breaker.

The next easiest step is probably to measure the current used by the circuit.  The easiest place to do that is in the breaker box with a clamp on current meter.  This would reveal whether the refrigerator is consuming more power than it is rated for&mdash;if you compare the reading with the ratings plate on the fridge.  If it is too high, first make sure someone hasn't violated the prime rule of a fridge circuit:  it should be the only device on the circuit.