What are some of the easiest mounts to get?

[Enkidoh]

If you level CNJ to 30 upon completing the level 30 class quest for it you can then do a very quick and easy quest from the CNJ's Guild in Gridania that rewards a Unicorn mount. Note that you cannot be WHM for this quest, you must be CNJ (so unequip the WHM Soul Crystal if you have it temporarily for it).

Near the end of Heavensward's story you also gain a certain dragon mount as part of the main scenario (the identity of which is a spoiler so I won't spoil it for you), so that is another one. Likewise later in Stormblood's main scenario you also get another certain mount upon completing a particular dungeon.

You can also buy mounts on the Mogstation, some of which are from past seasonal events (like the witch's broom), and some unique for the Mogstation (like Cloud's Fenrir motorcycle from FFVII Advent Children). You can also get mounts from the Collector's Editions of ARR and the expansions, which is available as a 'digital upgrade' on the Mogstation.

[Iscah]

...(the identity of which is a spoiler so I won't spoil it for you)

By the time you've given that much detail, it might as well be a spoiler by deduction.

"You'll get a mount from the MSQ near the end of Heavensward." That's all you need to say if you don't want to spoil things. Don't tell them what kind of mount it is.