Hi! Automatic pet feeding - who knew that would be such a hot application of littlebits?
As far as the build around the bits, I've gotta make a big plug for Legos. If you have access to even a small set of them, you're likely to be able to come up with a sturdy, reliable setup. For my build, I was lucky enough to have a budding engineer who love, love, loves building with Lego. He had built a conveyor belt out of pure love for making a conveyor belt.
But thinking about it critically, starting from nothing, your biggest goal for parts should probably be axels, wheels, and possibly gears. Turny pieces seem to be the keystone to many of the pet feeder builds.
After that, think about the food type you're delivering to your theoretical cat. Obviously, you'll need a different dispenser for wet food than for dry. If you can, I'd skew toward dry food, as it seems way easier to dispense pellets than anything else.
As far as the littlebits go, I found the cloudbit + motor combo to be a bit fiddly. I sometimes got perfect action - hold the cloud control button for 3 seconds = 3 seconds of spin. Sometimes I got continuous motor spin. I'm still tracking down that behavior. My version 2 may incorporate the servo instead, as I'm looking at a more fool-proof way to not dump a week's worth of fish food in the tank by accident/misfire. I'm also going to see if I can get a RaspberryPi-powered webcam set up so I can see whether the feeder is working while away. A constant process.
Let me know if you have any specific questions. Have fun solving your quest!