Leave right now, or in June at the latest!

On our last cross-country ride (by car with the family), I was surprised to see that Idaho and Montana were really really hot in July. On the Pacific Coast, however, you'll be OK all Summer.

BTW, if you go now, rather than in September, you'll enjoy more sunlight. Just make sure mountain passes will be clear of snow, especially if you plan "detours" in higher altitudes.

As for the Southern section (L.A. or San Diego to Texas), do it as late as you can, while avoiding snow. Should that be October or November?