Chain skip happening seemingly at random with a new chain
Pedaling while changing gears in important to avoid damage to derailleur and chain.  Gear shifting proper technique by Sheldon Brown.
Shift the gears so that chain is on the middle chain ring(at the pedals) and approximately on the middle gear on the cassette. Verify that the derailer with the chain is vertical, if not adjust accordingly. 
Also, it is possible that chain may be skipping if the derailleur is slightly bent - in this case bring your bike in to a reputed local bike shop(LBS) and have it checked out. It may save your derailleur from further damage and possible need to replace it.
Changing to an easier gear just before you climb and spinning pedals (lower effort but slightly higher pedal cadence), instead of mashing on a tougher gear (applying greater force and lower pedal cadence) helps avoid chain skipping.