Really it depends on what you consider a rock and the surface.

If you consider meteorites to be rocks, some meteorites that fall to earth are older than our solar system, so you’d have to find those, and they are pretty rare. I’m pretty sure we have some in a few museums, and if you got the right degree I bet they’d let you touch it.

If you consider only rocks from earth, the oldest rock on the surface somewhere is probably on the surface of the moon, as it is now very commonly believed that both the moon and Earth formed via a glancing collision of proto-Earth with a Mars-sized protoplanet named Theia. Since the moon and Earth are made of the same stuff, there are plenty of rocks on the moon from this time that are not subject to erosion like rocks on Earth. We also have some samples here on Earth that you don’t even need a degree to touch, let alone a spacesuit. If you consider all rocks that were technically part of earth, this might be your next best bet.

If you consider only rocks that formed on Earth as its own, separate astronomical body AND are on the surface of other Earth rocks, you’d probably have to actually go to the moon. Because there were so many large impacts from other astronomical bodies during the early solar system, many fragments of Earth rock were thrown back out into outer space, many of which landed on the moon since it is the next closest large object. Most rocks that did form on the early Earth have since been recycled by Earth’s geologic processes, which do not exist on the moon. So while Earth was still so hot that nearly the entire surface was molten rock with just a few floating islands of solid rock, some impact likely knocked some of that solid Earth rock onto the surface of the juuuuuuuust cool enough surface of the moon to now be the oldest remaining Earth rock. This, unfortunately, you’d need to do extensive exploring and sampling of the moon to find. Not a great option. But MAYBE one of these can be knocked back onto Earth by an impact event that launches it back from the moon, and I am not joking. We have meteorites from Mars that we have collected on Earth, so this is not impossible.

If you only consider rocks that formed on Earth, are currently on the surface of Earth, and have never left Earth since its formation, then you are likely correct, and it’s in the Northwest Territories. It’s an ancient core material of one of the oldest even mountain ranges that formed on Earth, and it’s called the Acasta Gneiss Complex. This rock formation is just about 4 billion years old. Our solar system is only 4.6 billion years old. These rocks you can touch as much as you want, and they sit freely on the surface!

(Not a geologist warning)

Not the oldest, but a neat one: Banded iron formations. Probably formed because early life evolved photosynthesis, releasing large amounts of oxygen which precipitating iron out of the early oceans. Also "extinct" because the conditions for it to form probably will never exist on Earth again.

Stromatolites are somewhat similar. Refers to both current day living guys, and their ancient fossil ancestors. Both form layered mats and are some of the earliest fossils. I think there's earlier fossils but they're a good mix of old and decently large.