It’s really up to you when you decide to move to a new city according to your budget and resources. The best time to move to Los Angeles is during the winter months because during the summer months the hotel rates, sublet rates and temporary housing rates are higher because it is tourist season here. So anytime between October and March is an ideal time to move to Los Angeles.
If you are moving to other cities, it is a good idea to take into account the weather. You won’t be able to explore a city if you are buried under a foot of snow and if you decide to move to a city during the winter you will become depressed by staying inside all of the time.