It surprises most people to learn that 'paper' cups are neither 100% paper nor recyclable - or even compostable. If you think about it, a pure paper cup would soon fall apart when filled with liquid.The thing is that most paper cups are designed for single use and because they typically hold hot or cold fluids, they are generally made of paper with a very fine plastic coating. Because of the plastic coating, the cups cannot be recycled as paper and cannot go into your recycle bin. They have to go into your garbage bin.If your cup is PLA lined, it is technically compostable, but unless you have access to a commercial composting facility, this is a purely academic.
Paper cups are particularly deceptive little beasts. Beyond the fact that they can't be recycled or composted easily is the matter of how they are manufactured. Paper cups are generally made from renewably sourced wood, but the cocktail of inorganic chemicals used and water effluents produced in their manufacture means that paper cups may actually use more non renewable resources than polystyrene cups during manufacture.
PLA plastic lining is typically made from corn or a similar food that is high in starch. It also breaks down without leaving toxic residues, unless you have an issue with the fact that the corn used is generally GMO.