... I think I'll have the steak, ...
Tamura uses "I will"+ infinitive because he has just now made the decision to have the steak.
Other examples are:
I'm feeling sleepy, I think I'll have a cup of coffee.
I have some free time this weekend, so I think I'll go hiking.
... I think I'll have the steak, ...
We use "the" to talk about a particular countable object. In this case, Tamura has chosen one particular countable food item from the menu.
"Would you like" is used when making a polite offer.
Other examples are:
Would you like another beer?
Would you like something to eat?
Would you like salad or vegetables ...?
Be careful: when you are asked a question using "or", remember that the speaker is asking you to make a choice. This is NOT a yes/no-answer question.
... how would you like your steak ?
This expression is asking how long Tamura would like his steak to be cooked. There are three standard answers to this question:
- rare
- medium
- well done
I'd like a glass of red wine, please.
You can use "I want" in the same way as "I'd like" in this situation, however "I'd like" is a more polite expression.
Other examples are:
I'd like some water, please.
I'd like the chicken, please.