I've got a little wet bag (zip up cloth bag that's lined with laminate so its water proof) and you just throw them in there and zip it up.
For storing - apparently the more airflow the less the smell .. A laundry hamper style container is good. I've got a dog that would get into it all so I've got a bigger version of the wetbag to throw into a pail so poochy can't sift through and 'sample'. Also rinse nappies that have been used overnight to reduce strong wee smells.
Oh and kristyj!! Such a cute pic! That nose! Those lips!! Eeeee!!!