If there's some ability that states "increase your fly speed by 60 feet" or some odd other amount, but I don't have a fly speed and gain said ability, would I gain a fly speed of 60 or would I gain nothing?
No, it doesn't grant you a fly speed, it just increases it if you already have it. If you were to gain a flight speed (even a temporary one, for instance by being targeted with a Fly spell) then your bonus to fly speed would apply on that.
most abilities I've seen that are like this, say something like
"grants fly speed 30 (average) or +10 if you already have a fly speed"
So, by the RAW you wouldn't gain any benefit from said ability if you don't actually possess a fly speed. Think of it this way, if something said "increase your damage with swords by 3" it wouldn't actually do anything for you unless you had a sword and it certainly wouldn't cause you to spontaneously gain a sword that does 3 dmg.
What ability are you seeing this language on?
Nothing, I was just curious.
Null + 60 is still Null :D