・Why don't we go to ○○ next week?
来週○○に行かない？
・Do you want to go to ○○ with me next week?
来週私と一緒に○○に行かない？
"Why don't we 〜?" は「(一緒に)〜[しませんか？](https://eikaiwa.dmm.com/uknow/questions/54280/)」と誘う時のフレーズです。日常でもカジュアルによく使われます。
"want to 〜" は「[〜したい](https://eikaiwa.dmm.com/uknow/questions/10290/)(と思う)」という表現で、ここでは「〜に行きたいですか？」→「もし行きたいのであれば、一緒にどうですか？」と暗に誘っています。こちらもよく使われる言い方です。
[来週](https://eikaiwa.dmm.com/uknow/questions/132268/)XXへ行きません？
他の回答者さんもご紹介されている通り、Do you want to も使います。
例
Do you wanna see a concert on Saturday?
土曜日[コンサート](https://eikaiwa.dmm.com/uknow/questions/63398/)観に行かない？
Would you mind going with me to the shopping mall next week?
Hey, I am planning a road trip for next week. It would be nice if you can join me.
Hey, You keen on joining me for supper next week
>Would you mind going with me to the shopping mall next week?
*would you mind= (verb) be annoyed...
>Hey, I am planning a road trip for next week. It would be nice if you can join me.
By saying you are planning is indicating that you are thinking about it. "it would be nice" is telling the person that you really want him/her to join you.
>Hey, You keen on joining me for supper next week
*Keen=interested.
Would you mind going with me to the shopping mall next week?
*would you mind= イライラさせるという意味です。
Hey, I am planning a road trip for next week. It would be nice if you can join me.
「you are planning」ということでそれについて考えていると言うことができます。
「it would be nice」とは、誰かに参加してほしいときに使う表現です。
Hey, You keen on joining me for supper next week
*Keen=興味があるという意味です。
Do you want to hang out sometime, maybe go to the movies?
I'm going to the pool next week, do you want to join?
We should go out to eat next week. - This is a suggestion, that way the person doesn't feel obligated.
Do you want to hang out sometime, maybe go to the movies next week? - You're inviting the person out next week, but also leaving room for them to suggest another date, if they can't go out next week.
I'm going to the pool next week, do you want to join? - By telling the person your exact plan, they know if they would like to go, sometimes people like to know where they are going and the things they will be doing there.
Here are some more examples:
I'm heading to the mall next week. Would you like to join?
I wanted to hangout next week, are you free?
Are you doing anything next week? Would you like to hangout?
We have not hung out in a while. Would you like to hangout next week?
I can't make it this week. Can we hangout next week instead?
Let's hang out next week, I have not done anything the entire summer.
I know you said you wanted to do something this week, but can we do something next week instead?
Would you mind doing something next week, instead of this week?
We should go out to eat next week.（来週ごはん食べに行きませんか？）
→これは提案です。相手の断りやすい言い方です。
Do you want to hang out sometime, maybe go to the movies next week?（今度遊びませんか、「来週映画を見に行く」とかどうですか？）
→これは「来週遊びませんか」と相手を誘っていますが、都合が悪い場合に相手が別の日を提案できるような聞き方をしています。
I'm going to the pool next week, do you want to join?（来週プールに行くんですが、一緒にどうですか？）
→具体的に何をするのか伝えると、相手は行くかどうかの判断をしやすいと思います。どこで何をするのかで行くかどうかを決めたいという人もいますね。
次のように言うこともできます。
I'm heading to the mall next week. Would you like to join?（来週モールに行くんだけど、一緒にどう？）
I wanted to hangout next week, are you free?（来週遊びたいなと思ったんだけど、空いている？）
Are you doing anything next week? Would you like to hangout?（来週は時間ありますか、一緒に遊びませんか？）
We have not hung out in a while. Would you like to hangout next week?（しばらく遊んでいませんよね。来週時間ありますか）
I can't make it this week. Can we hangout next week instead?（今週は駄目です。来週は時間ありますか）
Let's hang out next week, I have not done anything the entire summer.（来週遊びませんか。この夏は何もしていないので）
I know you said you wanted to do something this week, but can we do something next week instead?（今週何かしたいって言っていましたよね。それ来週では駄目ですか）
Would you mind doing something next week, instead of this week?（会うのは今週ではなく来週では駄目ですか）
1. I'm thinking of going to the beach next week. It would be so nice if I had some company.
2. I'm considering going to a concert next week but was thinking I might get lonely on my own.
3. There's a great restaurant I have to show you sometime. You would love the food!
All of these are indirect proposals. You are putting yourself in a position where your friend or significant other does not have to say 'Yes' or 'No'. and you do not have to worry about being rejected directly either. However there is considerable psychological pressure on your buddy to 'step up to the mark' and help you. You are really saying that you will feel lonely or alone in these situations - or that you want to share with your friend the delights of an amazing restaurant. How could anyone resist such clever persuasion?
I'm thinking of going on a trip next week, are you keen?
I have planed something for next week, would you be keen?
In both cases, you can use 'keen' in a casual manner. This is to ask the person if they would be interested in what you planning. You can use these expressions freely without feeling that you being too casual or too formal.
2. Do you want to be my date to the.....next week?
1. Can be used for a friend or significant other
2. Can be used for a friend (more suitable between females friends) in a playful way without implying/saying that you are dating or in a relationship with that person. Or it can be used for your significant other.
You say/ask :
'Would you like to come with me next week to ...?'
'Join me next for ...'
'Come with me next week to ...'
'I want you to go with me to ...'
このように言ったりお願いしたりすることができます。
'Would you like to come with me next week to...?'
（来週私と一緒に〜へ行きませんか？）
'Join me next for ...'
（次の〜に一緒に行こうよ。）
'Come with me next week to ...'
（来週〜に一緒に行こうよ。）
'I want you to go with me to ...'
（あなたと〜に一緒に行きたいです。）
I have something planned for next week, do you want to come with?
If you want to casually invite someone to go with you somewhere next week, you can say:
"Do you want to go out some time next week?"
"I have something planned for next week, do you want to come with?"
「来週一緒にどこか行かない？」とカジュアルに誘いたいなら、次のように言えます。
"Do you want to go out some time next week?"（来週一緒にどこか行きませんか）
"I have something planned for next week, do you want to come with?"（来週行こうと思っている所があるんですが、一緒にどうですか）
Do you want to go to ~ next week?
来週〜に行かない？
Do you want to go to ~ next week with me?
来週〜に一緒に行かない？
上記のように英語で表現することができます。
with me で「私と一緒に」を表すことができます。
お役に立てればうれしいです。
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