I really want to take an overnight train one way and a day train the other way (to see the country) and it seems that there are only night trains between Paris and Barcelona. Am I looking at the schedule wrong?
Sounds like you're looking at the wrong schedule. This is the train schedule website used by many here: http://reiseauskunft.bahn.de/bin/query.exe/en * * * Try it and see if you get different results.
There are a couple of day trains. The best, with only one change, seems to be leaving Paris Gare de Lyon at 11:20 (TGV 6209), arrive Perpignan 16:25, change trains, leaving at 16:54 (TLG 70), arriving Barcelona (Estacion de Francia) at 19:59.
Another good choice with only two connections would be Gare de Lyon-Perpignan-Port Bou-Barcelona Sants, departing at 07:18 and arriving at 17:09. You would have the whole evening ahead of you after you arrive. A faster trip but with one more connection would be Gare de Lyon-Montpellier-Perpignan-Port Bou-Barcelona Sants, departing at 08:20 and arriving at 17:39. You also have several morning departure choices for Barcelona-Paris.