Carrot: an orange coloured root vegetable from the parsley family and a good source of carotenoid nutrients including beta-carotene
Roast lamb isn’t the same without roast potatoes and carrots

Caret: a mark that indicates where text is to be inserted. In editing, it is ^ but on a computer screen it is usually >.
The fox jumped over ^log.

Carat (ct):a unit of weight used for gems and precious stones. It is a metric measurement based on 0.2 grams per carat.
A five carat opal weighs about one gram.

Karat (k): a measure of the purity of gold – pure gold being too soft for jewellery, it is mixed with other metals to make an alloy.
My necklace is 18k which means it contains 75% pure gold.

Telling them apart can be simplified. Carrot is the most commonly used version, and the only one likely to rot.

Caret is a very specific word that most people would rarely come across, especially as the word cursor is used more often in a computer sense.

As for carat and karat, carat refers to how much gemstone you have to carry (i.e. the weight) while a karat refers to the King’s gold!