Hey y'all. I'm going to be DMing a game (SKT) for the first time, and one of my players has never played before. She wants to be able to ride a flying creature around. I already told her that I wouldn't let her ride a dragon. Does anybody have any ideas for how I can do something like that for her?

Her character is a human ranger, but she may be interested in multiclassing in the future. 

Is she a beast master? A player of mine tossed around the idea of a gnome ranger so they could do the same thing, an idea I really like myself. If I’m not mistaken to use something as a mount the creature must be 1 size bigger than the rider and beast masters can only have animal companions up to size medium, which is the same size as a human. So you’ll have to homebrew to either allow them to ride creatures of the same size or have a companion that is size large, there might be a couple options here that aren’t too OP. The beast my player was interested in riding was a pteranodon.

if they aren’t a beast master and they want to ride stuff, charm magic or simply animal handling checks to try and tame a beast or otherwise should work, coupled with checks to control the beast in flight and to stay on while being buffeted with attacks during combat.

I understand that she's a Human, but if she's willing, she could go Halfling (or any other Small race) and fly around on a Pteranadon. Then, she could go Beastmaster and have the Pteranadon be even cooler!

Otherwise, ¯\_(ツ)_/¯

In SKT I believe there is a npc who can train characters to ride hippogriffs, and another npc who can train characters to ride griffons. Alternatively you could take one of the giant vultures the evil air cultists ride or the giant ravens the raven uthgardt tribe ride.

It's important to recognize that a mount is different than a beast companion. A beastmaster gains benefits of the beast and the animal takes a much more active "adventuring" role. A mount would be (in my understanding or interpretation) a passive creature used for transport, or to ride in combat (separate mounted combat rules apply) but otherwise doesn't do anything on its own such as make separate attacks. Unless, of course, it's doing something on its own that makes sense for its own self-preservation, such as flying away if something attacks it.

Having a mount can be a benefit as much as a burden/liability. Anybody regardless of class or species can have a mount (indeed! Most whole parties at some point in most campagnas are likely to need their own horses for longer distance travel!). 

A mount is simply that: a mountable creature used for transportation. Doesn't matter per the rules what type of movement (walking, flying, swimming), nor limit any creature type or size (aside from needing to be at least one size larger). The player handbook even gives the Pegasus or Griffon as examples. It's really about what would make sense or be fun. Just because a mount happens to fly doesn't really change it from being anything different than treating it like you would a horse.

I would encourage to let the new person have their flying mount of a reasonable choice, place the burden of reading/knowing the rules of having/using a mount, reach an agreement that they'll be responsible for its upkeep care, stabling it when necessary, animal handling, etc. I'd recommend for simplicity's sake to limit the options to cr 2 or lower, beast preferred but openminded to other types (celestial, monstrosity). A creature with higher Intelligence should manifest a more noticeable personality given by the DM (good, bad, stubborn etc.) because even a run of the mill Riding horse can be very temperamental. If there's an interest in a more exotic creature, that could open up an opportunity for said player to spend part of a session seeking out an exotic animal seller or trapping/taming one, or getting the services of a specific creature, like a Pegasus, due to rescuing it or something similar.

If I was told I could have a flying mount, I'd go Giant fly all the waymonster