I have toured in trainers (sneakers) and platform pedals with toe clips, but I don't think it's a good idea for most people. You want a stiff sole in contact with the pedal. Some shoes and sandals can accept a cleat and still allow you to get off the bike and walk almost normally -- look at mountain bike shoes, and sandals such as those by Keen or Shimano. Or you can use road biking shoes and carry a lightweight second set of shoes/sandals to slip on when you get off the bike for more than a couple minutes.
Likewise, I've done gym shorts, but it really is wiser to choose cycling shorts. Some of those, too, are a compromise with off-the-bike style. Some of them have a semi-baggy exterior so they don't look too goofy off the bike.
Small note: From L.A. to San Francisco, you will probably have a headwind in that direction most of the time. It's not that bad, not a deal-killer. Just don't be too surprised if you spend 4-5 days being annoyed by it. Your itinerary is going to take you through a bunch of different weather.
Read up on the gear lists at this site, at www.crazyguyonabike.com
, and in books like the Sierra Club's guide to bike touring. Then read up on "ultralight bicycle touring" (Google it). Then synthesize all that to fit your standards for weight, convenience, risk and cost. There is no "right" gear list for everybody.
Enjoy the trip!