There isn’t a true answer to this. There are distinct seasons, but you have to decide which is best for you.
The first is summer. This is the busiest season. But there’s a reason it's so busy. For families with kids, this is when school is out and you have more time. For everyone else, this is when the weather is at its best. There also aren’t any major holidays to work around.
While summer is a great time to move, it's difficult. You’ll have to schedule your move in advance and the price could be higher. Planning for a winter move means a simpler moving process.