Homestay questions

my homestay japanese mum asked me to go to karaoke together.
I'd like to accept because i love karaoke, but i don't know how does it works:
is she paying for me too? it will be embarassing if I pull out my wallet to pay and she wanted to pay for me, but it will be embarassing also if i am certain that she will pay for me and so i won't make a movie to pay...
And if she ask me to to togheter to Kamakura (for example), she is going to pay my ticket or i have to buy mine?
I understand that there are varous kind of family, but i'd like to know wich is the normality..

I'd say, always be prepared to pay for your ticket, meal, drinks, or whatsoever.

If your host family invites you to go out with them, then it would be it will be embarassing if I pull out my wallet to pay and she wanted to pay for me

I don't think so :) You can always go up to the cashier with the mom, and offer to pitch in. If she accepts it, fine. If she says no you don't have to, ask again, and if she still says no, say thank you and let her pay for the whole family, including you. For going out somewhere, having to buy tickets, again, when she goes to buy tickets, go with her.

If she'd paid for all of you for you a few of times, THEN the next turn comes around, be it lunch or whatsoever, tell her it's your turn now to pay. That would be nice reciprocity.

Whenever you accept an offer to participate in an activity (travel, entertainment, dining), you should always be prepared to pay your own way.

The problem is, if you have a homestay host that is always eager to take you around you may start to feel pressure if, like most students, you're on a tight budget.

It's hard to say to your host "I'm broke". But even if that's not something you're comfortable saying flat out, you do need to convey that you're living on a budget to avoid uncomfortable situations later.

Politely decline by saying you're saving up for a trip (今旅行計画のために貯金してるから、落Zがない。。。） or something similar.

Also, I think it's fine if your host offers to pay for you and you accept. Show your appreciation at a given time (birthday, holiday, at the end of your stay).

Lastly, perfectly normal for older people to pay for younger people. There is a natural cycle of "pay it forward": someone did for me, now I do for you, and someday you will do for someone else. Just remember never to assume and never take anything for granted.

thank you for your answer, i have one more question: i have yet paid my homestay (i'm going next week) and breakfast and dinner are inclusive. If family go out to dinner, they HAVE to pay for me right? because i'm a little bit in a budget and surely i can't affort paying for dinner too all the times.
I've already paid 2500 yen a day and this should cover a dinner too..

That's a question about which you should confirm with your homestay program.

If the contract requires you to pay for breakfast and dinner, which you have, it would be unfair to expect you to pay for meals eaten out with your family.

2500/day is quite an amount... I hope they feed you well.

Make sure the rules of your homestay situation are 100% understood by you and the family, otherwise there could be problems in the future.