Put one of these on the stop valve (hot) under the sink: (3/8" female compression x 3/8" male compression x 3/8" male compression "quick tee")
Use a 3/8" braided stainless dishwasher connector to feed the DW off the side branch of the quick tee.
Put the DW discharge hose into the same standpipe/trap as the washing machine.
The utility sink faucet is 3/4" hose thread. You'll need adaptors to go from 3/4" female hose thread to 1/4" (maybe 3/8" - I can't remember off-hand) male compression. The whole assembly is screwed onto the faucet spout. A replacement spray hose kit can be screwed to the male compression adaptor.