I've had a shredder in the past, electric. I personally find that my mower does a better job for anything up to about half inch thick, just keep piling it and going over it.
You can compost it or use it straight away for mulch. I don't think people in this country really appreceate the benifits of plenty of mulch, improves the soil, helps to choke weeds and reduces the need to water.
Brian