Electrical circuit with too many lights and receptacles on it
You will need at least 1 new circuit, and to break the circuit and install that new 15amp circuit (I am guessing the wiring is 14 awg). You can get things to work by removing some of the load but you probably have figured that out.

If you have room adding a breaker is not that hard but you will need to do some reading and ask more questions.

You cannot use 20 amp breakers on 14awg wire--you have to stay with 15 amp breakers unless the wiring is 12awg. 

I doubt you have a short just a simple overload, so you will need to run a new circuit. 

To prove it’s not shorted, turn the lights off, unplug things, then reset the breaker. Now turn each room on then off and start plugging things in. If a single device like a electric heater is on the circuit, just leave that unplugged and everything else will probably work.