Farm fresh carrots are divine all year round, but they become exceptionally sweet after the first frost of the season which triggers them to convert starch to sugar.
If you're not eating them right away, cut off the stems and store separate from roots. Put each in the fridge in a sealed plastic bag.
Carrots will last 1-3 months depending on variety.
Tops will last up to 7 days.
An easy way to use up a carrot or two is to shave/grate onto a salad.