Why are they considered "clipless" when they clip into the pedals? That seems confusing.
The pedals are called "clipless" because they have no toe CLIPS to slide your shoe into. Long ago bicycle racers used toe clips on the front of the pedals (steel, not plastic) and had cleats on the bottom of their racing shoes. These cleats had a slot in them to slide over the back cage of the pedal. Shoes had wooden soles so the cleat could be nailed onto the sole. Plastic soles and screwed on cleats came along in the 70s and 80s.
Here are pictures of what toe clips and cleats used to look like. These are modern plastic sole shoes with screwed on cleats. At the bottom of the page is a picture of leather soled shoes and nailed on steel cleats.http://www.yellowjersey.org/tocleat.html
Here are original racing shoes with wood soles and nailed on cleats.https://www.pinterest.com/pin/314055773984550351/
And this is a picture of what people mean when they say toe clips today. Plastic hoops on the front of the pedal, nylon straps, and sneakers.http://www.bywayofbicycle.com/wp-content/uploads/2013/03/toe-clips.jpg
As for shoes and pedals to use on a cross country ride, get some SPD sandals. Shimano makes good sandals. And some cheap SPD pedals. Shimano makes some cheap models for about $30 from Nashbar.com You're all set.