きのうは あめでした/ it was rain.

あめ

means "rain" as noun.

あめでした

means "was rain.". "でした" is the past tense form of "です". "あめ" is a noun, so you can just connect with "です" or "でした" to make sentence.

きのう

means "yesterday".

きのうは　あめでした

means "it was rain.".

さむい

means "cold", which is i-adjective.

さむいです

when you make sentence in present tense with i-adjective. it's ok to connect simply with "です". However, when you make it in past, you need to do differently, it's ....

さむい＋かったです。

さむい　　かったです。

さむかったです。

remove "い" from the i-adjective, and put "かったです" after it. "さむかったです" means "It was cold.".

きのうは　さむかったです。

means "Yesterday, it was cold."