I have written a children's book about dragons and the things they do in Texas. Do I need to ask i.e. the Lubbock Arboretum if it's okay that I mention them in my story? So my question is: When do I need to ask permission to use a location? Or do I even need to? Thank you!
If by a "place" you mean a city, state, country, etc, then no. No one owns the name "Texas" that you could get permission from if you tried.
If by a "place" you mean an organization, like the headquarters of a company, things are a little more complicated.
There are two potential issues: libel and trademark.
If you portray a real organization as doing something illegal or scandalous, they could sue you for libel. Whether they could win depends on a variety of factors. 1. Did they really do what you said they did? If so, in the US it's case closed, truth is an absolute defense against libel. In other countries, like the UK, that's not the case. 2. Would a reasonable person believe you were saying this organization really did this, or is it obviously fiction? That can be tricky.
In general, the purpose of trademark law is to protect buyer and seller against the buyer being tricked into thinking this was the "real" product when really it's a copy. Like, if you made your own soft drink and called it "Coca Cola", and sold it in cans that resembled real Coca Cola cans, the Coke company could sue you and would almost surely win. You're tricking people into buying your product by making them think it's Coca Cola. But if you write a book and in it you say, "Bob drank a Coke", no sane person is going to confuse your book with a soft drink. No one is going to think that because your book includes the word "Coke" that therefore it is a soft drink.
Yes, sometimes this can get ambiguous. If your whole book is about Coca Cola and you have a picture of a Coke can on the cover, etc, Coke might be able to argue that you're giving people the impression that your book is an "official" Coca Cola product. They can afford better lawyers than you so maybe they'd win that one.
As the author, you don't need to be concerned with obtaining permission to use the name of a city, building, or whatever.
This is a concern for a publisher -- which might be you if you self-publish. The situation that would matter to a publisher is if the author is using a trade-marked name without permission.
While it is very likely that Lubbock Arboretum has trade-marked its name -- to protect its brand if they sell merch, this protection is very specific as to the form. As in, Lubbock Arboretum on a hat or a tee-shirt or hoody. Maybe with some iconography. Since they have pay to register and maintain the trademark on each and every form.
Consider the Louvre. If you were to make and sell unlicensed "I <3 the Louvre" totes, you could find yourself on the wrong end of an infringement lawsuit, but you can mention the Louvre in your novel or short story without concern for infringement.
If your story brought discredit on the Louvre or the Lubbock Arboretum in a way that a reasonable person might believe is true, then you could get sued. But that wouldn't be for infringement but for brand diminishment -- egregiously hurting their earning potential with false and misleading statements.
I would absolutely avoid depicting any non-public place, non-public building or private company in any way differently from the most positive things anyone would expect to happen there in reality.
That is, if your story takes place at a private Arboretum and your protagonist goes there to look at the trees and meets a nice person there, no problem.
But if in your story that Arboretum is a place where dragons live, that is potentionally problematic. Because the owner of the Arboretum place might not want his business to be associated with dragons, no matter how positive you think your portrayal of the place was.
Photographers have to get a location release by the owner of a place if they want to commercially use photos they took of that location, and this is no different: You are commercially using someones private property, so absolutely avoid anything that can be construed as being less than positive.
Public places, state owned property, countries, and so on you can do whatever you want with.