と marks the end of what was thought or said, and everything before it goes into plain form no matter how polite the sentence is. Embedded questions with かどうか work the same way.
Also known asto omoimasuto iimasuと思いますquotation toka dou ka
Plain-form clause + と 思います = “I think that …”.
あした あめが ふると おもいます。
I think it will rain tomorrow.
The content of your thought is a plain-form clause marked by と before 思います. Use this for opinions and guesses.
In the textbooksMinna I, lesson 21Genki I, lesson 8
と marks a quote before 言います. Direct quotes use 「」.
たなかさんは 「あした やすみます」と いいました。
Mr Tanaka said, “I'll take tomorrow off.”
と quotes speech or thought before 言います. A direct quote keeps the original wording in 「」; an indirect quote uses the plain form before と.
In the textbooksMinna I, lesson 21Genki I, lesson 8
でしょう = a guess (“probably”) or, with rising tone, “…right?”.
あした はれるでしょう。
It will probably be sunny tomorrow.
でしょう expresses probability. Said with rising intonation (でしょう？) it seeks the listener's agreement: “…isn't it?”.
In the textbooksMinna I, lesson 21Genki I, lesson 12
と書いてあります = “it says …”; 〜という意味です = “it means …”.
いりぐちに 「たちいりきんし」と かいてあります。
It says “No Entry” at the entrance.
と quotes the words written or said; 書いてあります reports what a sign/notice says. 〜は どういう 意味ですか asks a meaning; 〜という 意味です gives it.
In the textbooksMinna II, lesson 33
Embed a yes/no question with 〜かどうか, or a wh-question with 疑問詞 + か.
やまださんが くるかどうか わかりません。
I don't know whether Mr Yamada is coming.
A yes/no question becomes 〜かどうか (whether or not); a wh-question keeps its question word + か (どこで 買ったか). The embedded clause is in plain form.
In the textbooksMinna II, lesson 40