If you are talking just the appliance moving, your most professional and likely least damage option is Movers (your Search the List category to find well-rated and reviewed vendors for this) - regular moving companies - typically about $150 minimum charge for a couple item in-house move like this. Hauling is another Search the List category for this - the two guys with a truck type junk hauling companies.
Also, if you have some strong friends willing to help out for some pizza and beer, that is a common way to get it done - though a bit of liability if they hurt themselves, and more likely to dent appliances or walls.
Remember the washer and hoses will have water in them after the water faucets to the hoses are shut off - probably about a quart or so per hose and maybe that much again in bottom of machine in some cases, so drain them into a pan before moving - tilt to back to drain out.
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If you need new utility connections in the new location (electric, gas, dryer vent, water, drain line, etc) here are links to a bunch of previous questions about that:
https://answers.angieslist.com/How-co...
https://answers.angieslist.com/I-wash...
https://answers.angieslist.com/Reloca...
https://answers.angieslist.com/what-t...
https://answers.angieslist.com/I-plum...
https://answers.angieslist.com/How-I-...
https://answers.angieslist.com/How-mo...
https://answers.angieslist.com/Is-typ...
