Somehow I was lucky enough to hit something when going downhill. And now my tire looks like in the picture. So should I still drive with this tire or get it replace immediately? And to replace it, is it safe to drive to the tire shop, which is a few miles away?
Sidewall damage is a serious issue. Sidewall is the structural part of the tire, and damage to it can result in dangerous consequences.
Replace your tire immediately. If you know how to put the spare tire on (and there is a spare tire in your car), please do it now, and drive only to the tire shop. Spare tires are not designed for extended drives, so don't start driving around in the spare tire.
If you have driven more than a few miles already with the tire before looking at the damage, you may have enough data to indicate that it may survive to the tire shop. I would however prefer the spare tire approach in your case. If there's no spare tire, then that would be a difficult decision: to have your car towed with the broken tire off the ground, or to drive the car. You know more about the details in your case than I do, because you didn't indicate how many miles exactly you drove with the tire after the incident.
If the tire is holding air, you should be safe enough to take it to the tire shop to get it replaced, but replacement is exactly what you should do. The damage on the tire is of enough severity there's nothing else you can do with it. Feel happy in the fact you didn't have a blowout on the tire, especially while going downhill, as this could have caused catastrophic results.
As others in this thread have already indicated: yes, that damage warrants immediate replacement. Since I can't comment on your question whether you should change all 4 tyres: no, only the damaged one and the other one on the same axle. So if it's a front tyre, you change both front tyres, and if it's a rear, you replace both rears.
If the tyres have been replaced very recently, this may not be necessary. But both left and right tyres should be of the same brand and model, and have an even wear.
YES REPLACE ASAP... please listen to all the commenters ... they gave great advice. Yes, 250 miles might be right if you are idling down a STRAIGHT field of pillows.
Put it this way, would you take an elevator with a tearing cable to the 98th floor? Any type of structural damage is AN ISSUE. I had a much milder issue with my tire, and it separated from the base of the lower wall just taking the on-ramp to I-95. Inertia + centrifugal forces + GRAVITY are a disaster. That's why cars immediately depreciate when you leave the dealer. MY last point: if you had to ask the question in the first place, get it checked out. Period. I wish I had. I hope all goes well.