（よ）う・
行こう is a decision as you make it: "I'll go" to yourself, "let's go" to others. With と思う it tells someone what you are thinking of doing.
to yourself
All right, I'll go to bed early tonight.
telling someone
I think I'll go to bed early tonight.
in mind for a while
After I graduate from college, I am thinking of working abroad for a while.
The form
行く → 行こう 食べる → 食べよう する → しよう 来る → 来よう
Godan verbs (う-verbs) change the last kana to its お sound and add う. Ichidan verbs (る-verbs) change る to よう.
Telling someone
（よ）う + と思う／と思っている
Polite と思います, と思っています. The verb before と stays plain: not 行きましょうと思います.
| Verb | Like this |
|---|---|
| Godan | 書く → 書こう, 泳ぐ → 泳ごう, 話す → 話そう, 待つ → 待とう, 遊ぶ → 遊ぼう, 読む → 読もう, 帰る → 帰ろう, 買う → 買おう |
| Ichidan | 食べる → 食べよう, 見る → 見よう, 寝る → 寝よう |
| Irregular | する → しよう, 来る → 来よう |
Deciding: 行こう
By itself it is what people say as they make up their mind, mostly to themselves.
I'll check it once more tomorrow morning.
I'll read this book to the end.
With かな you haven't decided yet. どうしよう is for when you don't know what to do.
I wonder what I'll have for lunch.
Maybe I'll go home now.
My wallet's gone. What am I going to do?
Telling someone: 行こうと思います
To tell someone else, people add と思います: "I think I'll…". It is softer than announcing 行きます.
This place is crowded, so I think I'll go to another restaurant.
I think I'll be heading home now.
Not 行くと思います
English "I think I'll go" has a plain "go". In Japanese a plain verb before と思う is a guess, not a decision.
I think I'll go to tomorrow's party.
What you mean to do.
I think he'll go to tomorrow's party.
A guess, usually about someone else. About yourself it is "I'll probably go".
と思う or と思っている?
と思っています says the idea has been with you for a while: "I've been thinking of". Something you have only just decided takes と思います.
The rent is cheap too, so I think I'll move here.
Just decided.
I have been thinking of moving to this town for some time.
In mind for a while.
と思います is your own thought, so it can't report someone else's plan. That takes と思っている, usually with how you know.
My younger sister said that she was thinking of studying cooking after graduation.
Not going to
行こう has no everyday negative. For "I think I won't", people say ないでおこう ("I'll leave it") or use つもり: 買わないつもりです.
I'm thinking I won't buy any new clothes this month.
Close neighbours
ましょうN5
Said to other people, 行こう is "let's go": the casual 行きましょう.
Let's walk to the station together.
つもりだN5
You have made up your mind. ようと思っている says the same thing less firmly.
I intend to move to Osaka next year.
予定だN4
It is on the calendar: a date is set or a booking made.
I am planning to travel to Kyoto next month.
（よ）うとするN3
The same form with とする is "try to" or "be about to".
I tried to open the door, but it was locked and wouldn't open.
Check yourself
1Which one fits?
明日は
Choose an answer to see why.
2Which one says what you have decided to do yourself?
Choose an answer to see why.
3Which sentence sounds natural?
Choose an answer to see why.
More detail
ようと思ったが
In the past it usually comes with a "but": you meant to, and it didn't happen.
I meant to study abroad, but put it off for family reasons.
ようとは思わない
With は and a negative 思わない, the idea is ruled out: "I have no intention of".
I have no intention of forcing a conclusion.
More sentences
The rain has stopped, so I think I'll walk home.
It is already late, so I think I'll stop here for today.
I think I'll ask about this issue at the next meeting.
I am thinking of taking a Japanese history class next term.
When spring comes, I am thinking of going to Kyoto to see the cherry blossoms.
For my health, I'm thinking of walking every day from now on.