If you’re not familiar with the term “yarn barf”, it is defined as “the mess of tangled yarn you pull out of the middle of a centre-pull ball”. It’s such a common problem that there are many crocheters (and knitters!) who avoid the issue altogether and just work from the outside end of the yarn.
You can avoid yarn barf that way, for sure, but I’m not convinced it’s the best way. When you work from the outside end, the ball of yarn tends to bounce around and roll all over the floor. Either that, or you have to keep stopping to pull off the next bit of yarn, which really cuts into your crocheting speed!
With experience, though, you can often avoid yarn barf. With some yarns, it is easy to find the inside end as it rests near the outside. With other yarns, though…
Bernat Satin is my most favourite yarn in the world, but it’s not one of those great “oh, here’s the end just sitting here!” kind of yarns.
Here is how I find the inside end (and what I do with it!) :
Find the end of the ball that has the outside end tucked into it:

Pull that end out:

and wrap it around the outside, tucking in the end (you may need to unravel the ball a little bit to do this) :

This step is key, for if you don’t get the outside-end out of the way, it will almost certainly tangle up with the inside-end.
Flip the ball over so that you’re looking at the end the outside-end wasn’t tucked into. Using both hands (not just one as shown), carefully open up the hole, staying centered:

Using thumb and finger, reach into the hole and feel around for the end. The end will be roughly in the middle of the ball, but it is unlikely you will feel it on the first try (if ever, really…). Try, instead, to find a strand of the yarn that feels like it should be right in the middle of the ball:

Pull that strand out. If it resists, it’s probably the wrong strand, however, once you start pulling there usually is no going back, so you might as well see what the damage is:

Not bad, really. It’s not a *lot* of yarn, and better yet, it’s not tangled! This is usually what I get when finding the end this way (apart from the rare instances when I actually do find the end on the first try).
Anyhow, pull the bundle of yarn out until there is only one strand running into the ball. That means the centre end is on the outside now.
To find the end in a way that won’t tangle the yarn, start by grasping the yarn that is coming out of the ball at the point closest to the yarn barf.

With your other hand, pull the yarn away from the yarn barf, to lay in a neat pile (on your leg, or the chair beside you, or whatever makes sense for the amount of yarn you’re dealing with).
Keep making the neat little pile until you get to the end:

If the yarn is just too tangled, you may want to think seriously about just cutting it at this point. You can always detangle it later and use it in some kind of scrap blanket or freeform project. Barring that, the tangle of yarn can be used to stuff amigurumi. :P
If you’re going to use the yarn right away, you can simply just go ahead and do so.
If you’re not going to use it right away, or if you don’t have time to use up all of the yarn that came out in one sitting, then read on:
Wrap the yarn in a figure-8 pattern around two of your fingers. Which two fingers will depend on how much yarn is outside of the ball. Mine is just a little bit, so I did pointer and middle:

Make sure that the end is sticking out at all times. You don’t want to lose it now!
Take the yarn off of your fingers, and start wrapping around the middle of the figure-8, still making sure your end is sticking out at all times:

Keep going, if needed, in a diagonal direction:

And then in the other diagonal direction, if needed.
When there’s a bit left, wind sideways again, but don’t pull more yarn out of the main ball, instead, move closer to the end of the main ball as you wind:

Once you’re just a couple of inches away, tuck the little ball you just wound into the ball band, with the end sticking up. You may have to flatten the little ball a little bit:

You are now ready to start crocheting! Just pull on the end sticking out of the little ball. The big ball will stay put, and the yarn will pull freely from the little ball until it runs out, and should pull seamlessly from the big ball from then on. There may be some hang-ups, but there shouldn’t be any secondary tangles.
If you find that you consistently run into secondary tangles, then I suggest that when you’re winding the little ball, that you *do* pull extra yarn out of the main ball until it is pulling out freely.
Sometimes it is just unavoidable and the whole ball needs to be re-wound. This hasn’t happened very often to me since I settled on the above method of inside-end-finding, but it does still happen. You can re-wind the whole ball using the steps described above, but do the figure-8 part around your thumb and pinky finger, your hand splayed out as much as you can. Keep winding around the middle, then diagonally until you are done, and whatever you do, don’t lost the inside-end!!