Partial power outage after installing new light fixture
A ohm meter or continuity tester will help you troubleshoot the wiring. With the power off and the light switch on check each set of black / white wires you are looking for the one that is close to zero ohms or shows continuity, then turn the light switch off and the reading should show close to infinite or no continuity. This wire is the switch leg. 

The other 2 cables are your power and the run to another light or outlet. Now if you tie all 3 blacks together that side is done. Mark the white switch leg wire with black tape and connect to the black wire from the light. Connect the 2 remaining white wires together with the white from the light. 

Since you have a metal box the ground (bare copper) should be attached to the box It may be better to pigtail the ground if the fixture has a ground wire connect 1  ground to the box then connect the incoming ground and light fixture together. This should get you back up and running. 

With a direct short the breakers clear quite fast and don't  usually damage the wires. to reset a faulted circuit breaker turn / push it all the way to the off position than back on.