Should the word climate be capitalized?

No, the word "climate" is not capitalized unless it is the first word of a sentence or part of a proper noun.

Here are some examples:

Not capitalized:

* The climate is changing.

* The Earth's climate is warming.

Capitalized:

* Climate change is a serious problem.

* The Climate Action Summit is taking place in New York.

Important note: "Climate change" is usually written as two words, but it is increasingly common to see it written as one word, "climatechange".