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). Foil-lined paper, like cake pads, boxes, or cupcake liners, belongs in the garbage.
Consider carrying reusables with you for coffee and tea drinks and take-out food. Ask your local eateries about replacing disposables with reusables.