I'd like to check out today.

You can use "I want"+infinitive in the same way as "I'd like"+infinitive in this situation, however "I'd like" is a more polite expression.

Other examples are:

I'd like to speak to Mr. Shimizu.

I'd like to see the menu, please.

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How will you be paying?

The clerk is asking whether Mr. Fujiwara will be paying his bill by cash, travellers' checks, or credit card.

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Could you call a cab for me, please?

Could you" + infinitive is used in this expression to make a polite request. You can use "can you" in the same way, but "could you" is more polite. In the U.S. and Australia people usually say "cab", instead of "taxi".

Other examples are:

Could you repeat that, please?
Could you sign here, please?

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