you have about summed it up,pump the pistons partway out,a good scrubb up with brake cleaner and a tooth brush ,then dry off and push the pistons back in.
they should go back with thumb pressure,if not the caliper needs to be stripped and serviced,dont be tempted to use WD40 or similar products as this can damage the seals.
it is worth spending a bit of time cleaning the caliper properly to remove all of the baked on brake dust,get in all the nooks and crannys,the dust holds moisture and will promote corrosion.
if your are servicing at home it really is worth allowing extra time to give the bike a deep clean while the panels,tank etc are off.
bikes that are kept clean generally have a lot less corrosion.