The Paphos Archaeological Park and Tombs of the Kings are two great outdoor sites in Paphos that are definitely worth visiting if you're there. They're sizable park areas for walking around by the coast and checking out the ruins up close.
You can generally walk all over and even inside a lot of it including the tombs. The Archaeological Park has many preserved mosaics with a building built over them along with some that were left outside. I've seen such mosaics in museums but it's interesting to imagine the ancient houses over them in their original locations by the sea.