Several different kinds of aphids may infest carrots, with some of them capable of transmitting a plant virus. So it’s to your advantage to use last years’ experience as a warning: Keep an eye out for early infestations and deal with them promptly. 

As you already realize, home gardeners need to be vigilant in order to prevent potential problems before they blow out of proportion. (Your ability to do that was revealed when you found the white eggs on your spinach leaves; that’s really good detective work!)

Aphids can be particularly challenging because they multiply so rapidly during the growing season. Even if you squish them daily and/or blast them off with a harsh water spray, just one remaining aphid can re-populate the plant in short order; no male aphids needed.

As soon as aphids start feeding, they excrete honeydew, a sweet sugary liquid which attracts ants. So focus your management on the aphids. At the first sight, employ your preferred tactic: squish them; use a harsh water spray to force them off the foliage; or apply direct hits of insecticidal soap diluted according to label directions. Additional options include azadirachtin (neem extract) and kaolin clay.  Kaolin repels some insect pests when sprayed on foliage.