Definitely realizing this game needed more tutorializing, sorry!
For the coffee machine, pickup one of the cups beneath it and then click the machine (if it makes a sound, it worked).
To serve items, once you have an order (by clicking the register) place items on the tall counter with left click/E. It should be lit up. Serve by clicking the register again.
EDIT: Add creamer to coffee by holding creamer and clicking on a coffee once it's been placed on the counter.