I thought that I was going to have to spray them, so I started doing some research on what would be most effective and whether there were any alternatives to pesticides. It turns out that ladybugs (and ladybug larvae) are voracious predators of aphids. I remembered seeing some other bugs on the leaves, so I went back out and took another picture:
Score, ladybug larva! Literally within days, the aphids were gone. Ladybugs are one of the best bugs you can have in your garden. They don't eat plants, but attack all kinds of other insects that do eat plants. If you don't seem to have any around your garden, you might want to consider buying a shipment of larvae and releasing them.
