Plain forms and polite speech
Plain form does not mean dictionary form alone. It includes affirmative, negative, past, and non-past forms, and it also appears inside otherwise polite sentences.
Four basic plain forms
The basic plain forms of 読む are 読む, 読まない, 読んだ, and 読まなかった. Their polite counterparts are 読みます, 読みません, 読みました, and 読みませんでした. Politeness and time reference are separate choices.
Nouns and な-adjectives take だ in the plain affirmative non-past, as in 学生だ and 静かだ. An い-adjective ends with 高い, not 高いだ. Different word classes do not all follow a verb conjugation table.
Plain forms inside polite sentences
明日は雨が降ると思います。 means 'I think it will rain tomorrow.' 降る before と is plain, but 思います at the end is polite. A plain form inside a sentence does not make the whole sentence casual.
In 昨日買った本です。 ('It is the book I bought yesterday'), 買った modifies 本. Do not attach the polite form directly to the noun as 買いました本. Separate internal grammar from the overall style addressed to the listener.
Choose a style for the situation
With a new acquaintance or in an information-giving setting, です and ます offer a consistent starting style. Choice also depends on familiarity, roles, and situation; age alone does not determine whether plain speech is appropriate.
Read a longer explanation to check for unintended shifts in sentence-ending style. However, mechanically converting plain forms inside quotations or noun modifiers to polite forms can create grammatical errors.
Check: Must 行くと思います。 be casual speech because it contains a plain form?
Show the answer and why
No. 行く is the plain form connected to と, while the final 思います makes the sentence polite. Internal grammatical form and overall speech style are different matters.