The fill hose is the thin hose that runs from the wall valves to the back of the washer. It is one of the few washer jobs that belongs in your hands. The valves behind it, the drain, and anything inside the wall belong to someone else.
What you can do
Shut the hot and cold supply valves behind the machine. Place a bucket and towels under the connections. Unscrew the fill hose at the valve end and at the machine end. Look at the rubber washers inside each fitting. Replace any that are cracked, flattened, or missing. Thread the new hose on by hand, then tighten it snug with pliers and stop. Open the valves slowly. Watch both connections while the washer fills, then check them again a few minutes later.
Write the install year of the new hose on the hose itself or in your maintenance notes. That way the next swap has a date to work from.
What to hand off
Hand these to a plumber or appliance technician: a supply valve that drips, sticks, or will not close fully; a drain standpipe or drain hose that is connected to the wall; any water line work behind the wall; any step that needs the whole house shut off; and anything near the outlet or the machine's control board.
Questions before you start
If you are going to hire the valve work along with the hose, ask these questions first. Will the quote cover a new supply valve if the old one fails during the swap? Who shuts the house water off if a valve breaks, and how long does that usually take? Which fitting material will they use, and is it rated for the water pressure in your home? Will they test the drain once the washer is reconnected?
Check your work
After any hose swap, run a short fill cycle and look under the machine when the fill ends. Keep the hose out of pinch points behind the washer so it does not kink. Check the connections again at the next laundry day. A dry floor and a dry hose end are the result you are looking for.