NOTES TO THE DIALOGUE

Can I get a seat in the next plane to Paris?
在请求别人做某事的时候英语中常用 Can I ... ? （我可以……吗？）
同样的表达方式还有：Could I ... ? / May I ... ?
Could I borrow your bike? 我可以借用一下你的自行车吗？
May I borrow your bike? 我借用一下你的自行车行吗？
三者中Can I ... ? 常用于口语中，Could I...? 口气比较婉转，May I...? 是较正式的用法。

I booked last week.
book是“预订”的意思，它的名词形式是booking。
book a seat in the plane / make a flight booking 预订机票
book a seat at the theatre / make a theatre booking 预订戏票
book a room / make a room booking 预订房间

If there is a seat in economy, give me that.
a seat in economy（= a seat in economy class）经济舱的一个座位
first class 头等舱,例如：
I want to get a seat in economy (/ first) class. 我想乘坐经济（头等）舱。
还可以这样说（注意动词fly的用法）：I want to fly economy (/ first) class.
air fare 机票费

We'll get you a seat with another airline if we have to.
offers a first class service 提供一流服务
They'll get him a seat in a plane on another airline.
In order to make sure of your flight you had better make a reservation one week before. Also, when booking seats, make sure whether you want a seat in economy or first class and reconfirm your flight by telephone one or two days before departure.