Start with carrots that are in good condition, firm, well shaped, and free from cracks and crevices.
Reject the ones that are limp and obviously stale.
Carrots retain their texture best when stored in the vegetable section of the refrigerator.
Cook in small amounts of water, in the microwave with very little water or steam.
Try baking carrots this method of cooking tends to caramelize sugar and enhance the flavor.
Do not boil, A significant amount of Beta Carotene is lost through this method of cooking. Beta Carotene is a major component of carrots with many health benefits such as converting to Vitamin A and supporting eye sight. You should aim to preserve as much Beta Carotene as possible.
Ionie Ponde, MS, RD, LDN
B Food Savvy 
Reference:
Radhika Bongoni, Markus Stieger, Matthijs Dekker, Bea Steenbekkers, Ruud Verkerk, November 2014, Sensory and health properties of steamed and boiled carrots (Daucus carota ssp. sativus), Int J Food Sci Nutr, PMID: 24964285, DOI: 10.3109/09637486.2014.931360