To know if you're paper cup, plate, or to-go container is plastic-lined (e.g. poly-coated) on the inside, look for the shiny plastic coating. Wax-lined cups are quite uncommon though treated like non-polycoated paper (compostable).
Consider carrying reusables with you for coffee and tea drinks and take-out food. Ask your local eateries about replacing disposables with reusables.