Quick list if higher level things I go through, assuming I already have a place I'm moving into and I'm doing the move myself:
Do a change of address with USPS and date it for moving day.
Stop buying a lot of perishable items.
Buy boxes, packing tape, and packing paper. Wrapping and wadding packing paper is as effective and much cheaper than bubble wrap. Get a tape gun, they are worth it. Go with medium boxes for the most part. A couple large for clothes. A few smaller for books and heavy items.
Begin packing: Out of season clothes, books, items not necessary for everyday use like keepsakes, collectibles, half the kitchen items.
Clean as you go if you hope to get your full deposit back. Especially the bathroom and kitchen.
Prepare your furniture for transport using blankets and plastic wrap. Wrapping in giant strange wrap keeps blankets in place and doors and drawers closed.
The goal in the days before moving day is to get down to the bare essentials, the last things to pack. Pack your suitcase that you will be living out of during the transition and start transitioning.
Call and notify all utilities of your final day and arrange to have final bills paid or inform them of your new address. Also call or change your address online for banks, credit cards, magazine subscriptions, or whatever else you have associated to your address. Call ahead to start new utilities as needed.
Map your travel if you are moving states. Trucks are slow. The first night you will be beat from loading the truck and cleaning.
On moving day, whoever is helping you gets lunch. Order pizza or sandwiches. Pick the truck up before they arrive. Clean after they leave. Have help if your loading your car onto a trailer, but you must be done loading the truck first.
It's nice if you can arrange for your landlord or manager to stop by to pick up key and do a walk through at the end of day so you view the apartment together.