The order matters more than exact dates, which depend on how large the office is. This is the sequence that works even for a small office.
Set the dates: end of the old lease, availability of the new premises, any works. The gap between them decides whether you need storage.
Do the three-column inventory and measure the new floor plan.
Ask for quotes for the move, dismantling and, if needed, storage, with the list in hand.
Tell staff, clients and suppliers, and update the address wherever needed, including with public bodies: your accountant knows which notices apply to your company.
Arrange internet lines and networking for the new office: they're worth requesting among the first things.
Pack by room, with labels saying where each box goes in the new office.
On moving day keep one person at each office, with the list.
Afterwards, check the list: what arrived, what's in storage, what's left to dispose of.