DH and I did the 1 day visit on a business trip. Until then, we were more WDW people, and it was my first time being there since I was a kid. We have since been back several times. We did hoppers and were able to get a lot done in both parks. I would highly suggest the hoppers since it just takes a few minutes to go back and forth, between parks. I would also suggest not making any table service reservations, and just eating quick meals at counter service places. A table service meal will take up a lot of time, and since you only have one day, I would maximize what you can do. I think we spent about 13 hours in the park going non-stop at full speed on our 1 day trip and we got the majority of things in both parks done, but that was before Carsland and World of Color.
As far as what not to miss, that is really going to depend on what your kids and you like. For me, Indiana Jones and Pirates are can't miss at DL. Radiator Springs Racers and Soarin are can't miss at CA. It might be different for you. We love shows and for us, even only having that one day, the 40 minutes for the Aladdin show was well worth the time to us. It is a great show. In fact, we changed a trip from WDW, to get DD back to see that show. If you aren't show people, or aren't big on Aladdin, that may not be a must do for you when you only have 1 day. For us, with only 1 day, we would not invest the time for the nighttime entertainment like Fantasmic, or World of Color (although I absolutely love WOC maybe enought to consider it even on a 1 day trip). Some people might feel differently. What kinds of thing does your family enjoy? Which Disney movies are your favorites? That might help you get some better advice on things not to miss.
One suggestion is to take advantage of the single rider lines. Indy and California Screamin usually have them, and I'm not sure what others there might be. You will have to split up to ride, but it will help shorten your wait and cram more into your day!
