Am I understanding the PHB correctly in that if, during a Wild Magic Surge, you cast Fireball centered on yourself, you too are damaged? That's a death sentence for lower level casters, which is quite possible since this sorcerous origin is chosen at level 1.
Yes, that's correct. Fireball's AoE is a sphere. From the Area of Effect rules on Spheres:
A sphere’s point of origin is included in the sphere’s area of effect.
So yep, you're going to hit yourself, and yep, it's going to hurt.
As a level 1 Sorcerer, you'll probably have between 7 and 9 max hit points. The saving throw DC will probably be 13, and your Dex bonus will probably be +2 or +3, giving you about a 50/50 chance of success. If you fail your saving throw, the average damage of 28 will probably kill you instantly, under the massive damage rules. If you make your saving throw, or your race gives you fire resistance, you'll probably still be making death saving throws.
You may want to suggest to your party members that staying 20 feet away from you could be a good idea.
Yes, the Sorcerer is affected
From the description of Fireball:
Each creature in a 20-foot-radius sphere centered on that point (point of origin) must make a Dexterity saving throw...
I added the point of origin clarification. Since a wild-magic surged Fireball is centered on the sorcerer, and there is nothing indicating the sorcerer is exempted from the damage, they must make the Dex save, taking 8d6 fire on a fail, half on a success.