When I was a child, I was even afraid of PICTURES of spiders and other bugs. I was just sure they were going to come to life and crawl off the page right onto me. I should have become a horror-movie scriptwriter.
I am still not fond of them, but once I grew up there was nobody else to get rid of the nasty things for me!
It's true that spiders can be very beneficial. That doesn't mean they have to have lunch with me at the dining room table, but if they want to hang out in the back yard I'll leave them to it.
*However,* I also did some research to find out which eight-legged intruders I should *really* worry about. And you might want to do that as well. Once you're home, you might google your county extension office and see what you can find out. Once you know the one or two spiders that are really unpleasant, you can learn to treat the others with polite indifference.
You could even google that information wherever you vacation. (I would have asked the desk clerk.) The more you know, the less fearful you are, and the less fear you will pass on to your children.
I don't like cleaning up insect, um, remains after they've been swatted or squashed, so I have a very elementary system of insect removal: a large plastic cup and a 4X6 (or 5X7) index card. Pop the cup on top of the offending arachnid (or other bug), slowly slide the card across the mouth of the cup to contain the perpetrator, hold on to both ends of the cup, take the perpetrator elsewhere, dump it out into the bushes, and tell it to go bother somebody else. I get laughed at, but it works.