Polite and casual
Every Japanese sentence picks a register. です・ます (polite) with people you do not know well, older people and at work; plain form (casual) with friends, family, and inside most grammar. Both are correct Japanese; the question is who you are talking to.
The two forms
Verbs: 行きます / 行く, 食べます / 食べる. Nouns and な-adjectives: 学生です / 学生だ. い-adjectives: 高いです / 高い. Negative and past follow along: 行きません / 行かない, 行きました / 行った.
明日、学校に行きます。
Reading: あした、がっこうにいきます。
I am going to school tomorrow. (Polite.)
明日、学校に行く。
Reading: あした、がっこうにいく。
Going to school tomorrow. (Casual.)
寒いですね。 寒いね。
Reading: さむいですね。 さむいね。
Cold, isn't it. (Polite, then casual.)
Who gets which
Polite: strangers, shop staff, teachers, colleagues, anyone older or senior, and every first meeting. Casual: friends, family, children, and people who switched to casual with you first. Switching down too early reads as rude; staying polite too long reads as distant.
Plain form inside sentences
Even in a polite conversation the plain form appears inside clauses: before と思います (I think), と言いました, から (because), とき (when), and before nouns. Only the end of the sentence carries the politeness.
行くと思います。
Reading: いくとおもいます。
I think I'll go. (Plain 行く inside, polite ending.)
雨が降るから、傘を持っていきます。
Reading: あめがふるから、かさをもっていきます。
It will rain, so I'll take an umbrella.
昨日食べたラーメンは美味しかったです。
Reading: きのうたべたラーメンはおいしかったです。
The ramen I ate yesterday was good.
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