食事にいきませんかって英語でなんて言うの？

1.
Would you like to - allowing for refusal is always polite.
join me - you can use this if you dont want to directly call it a date.

2.
I believe it's more common to say "can I take you out", but it's more accurate to say "May i"
Can - used to express that ability to do something
*May - asking for permission (polite).

3.
Go for lunch - this is informal.
On me - this means that you will pay for the meal. Another option is "my treat".

Go out means to leave a room or building, especially in order to do something for entertainment. Whereas the second sentence which is a very straigh way to invite someone's to go and having lunch.

Coffee is a good place to start. By sound it means is this a good idea.

Why not...? This is a rhetorical question and you would not expect a direct answer to this! If your friend starts giving you a list of reasons why they should not have lunch with you, that would be terrible! This question would normally be answered something like: "Oh, that's a good idea! Let's do it!"
Do you fancy......? = Would you like...?
A "Do you fancy going to the cinema?"
B "Yes, I'd love to!"
How about...? = Would you like...?
A "How about eating out tonight?"
B "Sorry, I have to get home for my dog. He'll be needing a walk."

Could I take you. means to take them out to eat.

Would you like to go? asks them politely if they would like to go with you.

Hello Konan :-) I hope this helps!

A) Would you like to grab some dinner with me ? -
*Grab - have (Dinner with me)

B) Will you please join me for dinner tonight? - Direct question.

▪ Hey, when can I take you on a date?
▪ Hey, will you go out with me sometime for dinner or lunch?
▪ I would like to take you on a date if you don't mind.

Any of the above sentences can be used. They are all clear and polite. All sentences indicate that you would like to take him/her on a date or take him/her out for dinner/lunch.