Laundry does take a bit of getting used to. Your Sim has to take the laundry to the washing machine (load it), then turn on the washing machine (“wash laundry” interaction on the machine that only comes up when laundry is loaded in there); wait, then they have to take the laundry to a clothesline or dryer. Dryer works similar to the washing machine. Load and then Dry Laundry. Don’t forget to clean the lint try between loads of drying!
The option to “Unload and Put Laundry Away” should come up on the Dryer or Clothesline when dry. If in doubt hover the mouse over one of the lit up options available (I find that they don’t all always work, but one inevitably does); this should give you a tiny white pop up describing the state of the clothes.
Whew, hopefully something there helps?