This has been something that has bugged me ever since Paarthunax and Tsun threw around terms like "Doom-Driven" and "Fateful" willy nilly. Are the heroes of Elder Scrolls bound by fate, or are they able to forge their own?

There seems to be evidence for both sides. On the one hand, it seems like fate forces the heroes into the situation that makes them heroes. This usually takes the form of the Hero being a prisoner of some kind. That might be symbolic of being a prisoner of fate. Prisoners are forced to do things against their will. I doubt the Dragonborn asked for the job, and he/she certainly didn't ask to be led to the chopping block on trumped up charges.

On the other hand, it seems like they do have some control. The stars have a lot to do with fate and at least the Dragonborn can change signs at will using the Standing Stones. Plus, if the previous two PCs were Shezarrine, they would be fighting for free will. This implies some control over destiny.

I'm new at this so I was wondering if anyone might be able to help me out.

"Each Event is proceeded by prophecy. But without the hero, there is no Event." - Zurin Arctus

To a point, yes, they have a date with destiny towards the end of their journey to do whatever it is that needs to be done. Beyond that though...there isn't much else. How they go about accomplishing their tasks is completely up to them, assuming they know what said task is to begin with. Aside from the Dragonborn and the Neravarine at the players digression, a lot of the PC's really seem more bound by circumstance throughout, as opposed to it being some birthright.

As Dargor said, they are 'bound by fate' in so much as they are forced into a deciding role. But what decision (and not just conscious, 'failing to stop satan' is still making a decision on the cosmic scale) they make isn't predetermined, and up to them.

It's like being on a road. You can't leave it, and eventually it leads to a crossroad. You can't avoid the crossroad, but you can decide which option to take. That is the extent of destiny for a hero.