I forgot to buy the things that my wife had asked me to buy.
「妻に頼まれた物を買うのを忘れた」は次の言い方があります:
1) I forgot to buy what my wife had asked for.
まず、I forgot to buy は「買うのを忘れた」です。
what my wife had asked for は「妻に頼まれたもの」です。
こちらの言い方のほうが短くてシンプルです。
2)I forgot to buy the things that my wife had asked me to buy.
こちらはちょっと長くなりますが、より詳しく、the things that my wife had asked me to buy 「妻が買ってと頼んだ物」と説明しています。ちなみに一つの物なら thing で、いくつかのものなら things となります。
ご参考に。