Whether you move to a new apartment once a year or once a decade, I don't ever think you get really good at moving. It's always going to be a chaotic time. A whir of bubble wrap and packing tape. But there are small steps you can take towards ensuring a smooth move–most importantly, not repeating the same stressful mistakes from the last time around.
Not starting early.
Getting an early start on your packing and moving adventure doesn't lower the stress of a big move, but it does spread it way out. Packing nonessential items three or four weeks in advance of moving day means that, one, you have more time to dedicate to the things that have to be done last-minute, and two, you have time to sell the stuff you don't want to keep. Cash in your pocket is way better than hastily running to the trash or donation center because you just want it out of the house.
Not taking time off.
Moving is an ordeal, y'all. And suddenly you, a presumably humble and clever little thing, suddenly get shoved into the role of CEO and COO of Getting My Stuff to The New Place, Inc. That's a full time job, with all it's logistics and scheduling and auditing, and if your work life allows, you should absolutely take time off from your day job to focus on the move.
Packing away the necessities.
You're finally in the new place, sitting on top of boxes to enjoy the Chinese take-out that just arrived and discover that they've forgotten to give you forks. This wouldn't be a problem if you'd taken a few extra steps to pack a few broad essentials into one single "life" box. What goes in there? Bedding, toiletries, phone chargers and as many place settings as there are people in your household.