From their side, they are saving the hassle of breaking down and disposing boxes. So for best results, ask them the preferred pick up time and stick to it. (there's a fine line between "less work disposing of boxes" and "tripping over stuff taking up space all day". If people feel appreciated, they can be remarkably kind.)
Oftentimes, deliveries come on specific days- for any store you ask, plan ahead slightly to work with their schedule. Not every store has the room or space to help, but some will try if they can. (Trader Joe's, for example, tends not to waste space on frivolities like storage. Or parking.)
Moving to a new place is a lot! I hope you find good friends and neighbors when you get there. Taking time to build roots before the new job gets hectic can make a big difference to quality of life later.