I moved one street away in Central, requiring the truck and 3 guys for everything in my apartment except the fridge. Price was $2400.
When my wife (then girlfriend) and I moved in together we took our 2 apartment's worth of stuff (both on the same street) from Central to TKO for $5500.
The latter was a 1st floor walkup, but the former had a 50m+ through the parking to the service elevator.
High-floor walkup with furniture, expect some to even turn you down, others to charge per floor. Can't advise on costs, I've got no first hand experience there.
If you're looking to make it as cheap as possible, ask for boxes in advance and pack yourself, and also disassemble all furniture yourself.