'Me gustaría' is mostly used when you are expressing something that you would like to do.
For example : Me gustaría ir al cine. I would like to go to the movies.
'Quisiera' is used mostly when giving some sort of polite command or indirect order or even asking for permission. As in your example, 'Quisiera reservar una mesa...' You are expressing what you wish to be done for you, in other words, you are somehow giving a command. As opposed to Me gustaría ir al cine where you are simply expressing a wish or desire.
so "me gustaria reservar una mesa para esta noche" is me talking to my friend about our plans for tonight...
and "quisiera reservar una mesa para esta noche" is me talking to the hostess of the restaurant on the phone?
Since the verb ponerse also means to try on, I think "I would like to try on those red shoes" should also be accepted.
Why do you think it also means "to try on"? Checking several sources I didn't find that, but rather "probarse."
The robots voice sounded like it said "Hiciera" and I got really confused
Grammatically, 'quisiera" is past subjunctive. Probably it's best thought of here as an idiom, much like "would like" is. "Would like" isn't normally a conditional about liking, it's a polite way to say you want something, which is what "quisiera" also is.
wouldn't this be better translated as:
"I would like to put on those red shoes."
'ponerse' doesn't really mean 'to wear'
That is possibly the most garbled text-to-speech rendition to date. Sounds like "Hiciera porde ez zapatos ros." I'm living in Mexico; I hear Spanish daily, and if I have to use the slow speed to understand it after playing it at regular speed 4-5 times, that's not an adequate audio file.