It's almost never too early to begin planning your move when possible. An early start can pay off in many ways: you can sort through your belongings at your own pace, for example. Donated used clothing and other household goods can provide a tax deduction, and you can schedule a yard sale at your convenience for the other items you no longer need.
If you can schedule your move 'off-peak season,' from September through April when demand for moving services is lower, you may get a better price and find that movers can be more flexible in their pickup and delivery dates.
If you do need to move during the summer, try to schedule it mid-month and/or mid-week, since the busiest times for movers are weekends and end of the month; which may also mean a lower price.