Our portable Kenmore dishwasher has started draining so slow it leaves water that has to be drained using the drain/cancel button multiple times. I think something is clogged up but have no idea where to look/start.

Look in these manuals: https://dl.dropboxusercontent.com/u/8010163/Whirlpool_Kenmore%20Dishwashers.zip
You have one drain filter at the bottom of your tub where it is easy to see and pull out and clean. It should be cleaned almost every time you use the dishwasher.