Can I change the starting planet in the Expansion Planner so that it will help me identify the planets with resources that are closest to my new colony?
Not that I know of. However, I also do not believe that the Expansion Planner distance is to the homeworld. Rather, I believe it to be the distance to your nearest colony. At any rate, that was how it seemed to me the last time I looked at the distance metric.
Is there an easier way?
Open up the Galaxy Map by clicking on the galaxy icon next to the minimap, or by double-clicking on the minimap. Go to the top left corner, set the filter to 'resources,' then set the filter to the resource you want to find. All known sources for that resource will be highlighted in yellow; if you know where your colony is on the map, you can use this to find a nearby source relatively easily, assuming that such a source exists. Alternatively, you can use the expansion planner to filter for resource locations and look at the minimap in the expansion planner; the resource site you have selected is at the center of one of the crosshairs, and an available construction ship (if any) is at the center of the other crosshair. Change the planet selection to see which crosshair is which, and then use this to figure out where a nearby source for whatever you're looking for is.
Also, it's not exactly necessary to ensure a good supply of resources for all colonies, only for the ones that you want to be building ships and stations at. Not all colonies need to have defensive bases, starports, and nearby mines, especially if they're low-population colonies. Focus your mining efforts around the areas where you want to put a starport or other shipyard.