According to the rules with 〜によると

Grammar Explanation: According to with 〜によると and 〜によれば

Now that Mao is living at Hiro’s house, Hiro presents her with a list of rules. Before going to bed, Mao looks the list over.

真魚(まお):: 「ルールによると朝(あさ)ごはんは必(かなら)ず一緒(いっしょ)に食(た)べる」; “According to the rules, we must always eat breakfast together.”; 「夕飯(ゆうはん)もなるべく一緒(いっしょ)に」; “We should eat dinner together too, as often as possible.”; 「朝(あさ)は一緒(いっしょ)に家(いえ)を出(で)る」; “We leave the house together in the morning.”; 「近(ちか)くの神社(じんじゃ)にお参(まい)りしてから会社(かいしゃ)ないし学校(がっこう)へ行(い)く」; “After praying at the nearby shrine, we go to work or school.”

Key Points

1ルールによると introduces what the rules say

ルール is the source of the information. によると introduces the contents of the list Mao is reading.
The rules follow directly, without a reporting ending such as そうだ. によると can name a source on its own.

2Plain verb forms state the household rules

食(た)べる, 出(で)る, and 行(い)く are dictionary forms. In this list, they describe what Mao and Hiro are supposed to do regularly.
The sense of obligation comes from the rule-list context. These verb forms do not mean “must” by themselves.

3必(かなら)ず makes breakfast firm, while なるべく allows flexibility

必(かなら)ず means “without fail”, so breakfast together is a firm rule. なるべく means “as much as possible”, so dinner together depends on what they can manage.
夕飯(ゆうはん)も adds dinner with “too”. The line leaves out 食(た)べる, giving the fuller 夕飯(ゆうはん)もなるべく一緒(いっしょ)に食(た)べる.

4家(いえ)を出(で)る means “leave the house”

With 出(で)る, を marks the place being left. 朝(あさ)は sets the morning as the time frame, and 一緒(いっしょ)に says they leave together.

5お参(まい)りしてから puts the shrine visit first

お参(まい)りする means to visit a shrine or temple to pay respects or pray. Here, 近(ちか)くの神社(じんじゃ) identifies the nearby shrine.
してから combines the て-form of する with から. They pray at the shrine before heading to work or school.

6ないし joins the two destinations with “or”

会社(かいしゃ)ないし学校(がっこう) means “work or school”. ないし is a formal way to join alternatives, fitting the written list of rules.
へ行(い)く applies to either destination. In this scene, Hiro goes to work and Mao goes to school.

食べる (To eat): 食(た)べる means to eat or consume food.
夕飯 (Dinner): 夕飯(ゆうはん) means dinner or the evening meal. It sounds more domestic and conversational than 夕食.
家 (House): 家(いえ) is a building where people live.
出る (To go out): 出(で)る means to go out or leave. It is also used when something appears or is produced.
学校 (School): 学校(がっこう) means a school, where teachers provide organized instruction to students.
行く (To go): 行(い)く means to go or move toward a place away from the speaker.