"I'll show you to your room" or "I'll escort you to your room".
Which option among those two is better to use if you work as a hotel manager and want to take the new guest to his room and basically show everything around?

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Answer

I would use "show" as "escort" can have an authoritative connotation.

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Either one would be perfectly fine. My personal preference would be  "let me show you to your room."

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The polite one is number 2. Escort...
 This answer refers from oxford dictionary online

to go with a person orvehicle, especially to make certain that he, she, or it leaves or arrivessafely:

Several little boats escorted the sailing ship into theharbour.

Security guards escorted theintruders from the building.

The police escorted her to theairport, and made sure that she left the country.

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to go with someone and show them a place:

People on the tour will be escorted by an expert onarchaeology.

from oxford dictionary online.

Answer

'let me show you to your room' is friendly and informal.
escort is ok but it is often used by security guards and police etc.
for example, a judge in a court would say to the court bailiffs 'escort them out of the building'

Escort may on some rare occasions provoke a bad response from some people.

much depends on the manner and perceived friendliness of the person showing guests their room.