I have a character that I want to have live for 200 years. I've used the beginning to fill out his backstory with events that are critical to the story, including how he became immortal. How would I write a time skip ahead? I do have events kinda planned, like I want him to master the piano and master painting during these 200 years and get educated, but how would I fill up these giant periods of time?
Start with the present
It's common practice (for good reason) to introduce a long-lived character into the actual story you're actually interested in, and just gradually fill in what's necessary about the long history.
Less-defined history gives you flexibility
I'm not saying don't foreshadow - but a character with hundreds of years of only vague history can organically "grow backwards" (if you happen to be the kind of writer who writes organically). This is not so much an invitation to give radical shifts like, "yeah, he was a pacifist, but I just decided that he's always been a warrior and loves fighting." On the other hand, you might realize partway through a story that it would fit for a pacifist immortal to have acquired his pacifism in some horrible civil war. If you've left room to elaborate on his history, new ideas are easier to incorporate. Plus, one-off jokes of "I remember; I was there," become possible. And maybe he's not even lying that he was there. (But don't make him have been everywhere.)
Mystery is good
If you're dealing with a long-lived character, don't feel like you have to define everything. Does the character play the piano? Great! Does it matter where and when that skill was acquired? Only if it adds flavor now.
Also, don't cram all the excitement of who and what this character has been into the opening chapter, and then feel like you have to awkwardly skip a bunch of time. Again, start where the present story begins, and sprinkle a few interesting details about the character here and here, with the reader having plenty of room to imagine that there are many more interesting details which just haven't come up.
Defining a thing to death makes it dull.
When necessary, use flashbacks
When all else fails, have your exposition... later, when you've got the reader hooked. You don't have to put that scene of him learning technique from some great painter at the BEGINNING of a novel. Put it in after some other character remarks how his style resembles Monet's work.
Show the finished character and have them remember details of their story that are relevant to the current scene as and when. Nine Princes in Amber does this quite well, the protagonist has had a very long and eventful life, much longer than your story requires. He remembers incidents from it as certain situations bring them to the fore, like sword training when he picks up a blade in self defense, and meeting Vincent Van Gogh when something nasty strikes him as picturesque. These touches slowly build up into a rich backstory as the tale goes on and provide a framework for his current actions.
Tell What Happened Vaguely
Do not go on too much. Also, no one has a perfect memory, so he shouldn't either. For example, Joe practiced art for 2000 years but pretty much ignored the world around him. He knew of some wars and some countries falling, but he didn't pay too much attention, else he would be confused by the present day.
Start With Right Now
Don't tell about Joe in the past, tell about him now. Drop his backstory in bits and pieces, like, "Oh," Joe said. "You like my art? I've practiced for an eternity, almost."
Also, a website for handling immortal characters (for DnD, but still relevant): http://rollforfantasy.com/ideas/immortal-characters.php
If you want to write these scenes, you could always put them in past tense. For example, 100 years ago...piano, nest chapter 75 years ago... etc. or you could just have him playing the piano during the story and say he’s 200 years old. Your readers aren’t that stupid. They will assume he’s provincially learned how to play piano unless he totally sucks. Give the reader some credit. They understand how knowledge works