I was thinking about this the other day, and I thought it was probably past time to write something down about what I think Cheryl's capable of. I've mostly left it up to a matter of interpretation and mystery, so I'll stick the meat of it behind an LJ-cut if you want to remain ignorant.
As people who've played the first and third Silent Hills know, Cheryl Mason is the reincarnation of Alessa Gillespie, the girl who was born to be a vessel for Samael, the "god" of the Silent Hill cult. She lived an unspecified number of years as Alessa, six years as Cheryl Mason, and then another seventeen as Cheryl Mason, a.k.a. Heather Morris. My version of Cheryl remembers most of her three lives with relative clarity, although she's prone to a sort of disassociative disorder bordering on the point of having multiple personalities.
In her original incarnation, Alessa is often credited with the creation of the otherworld and the destruction in the streets of the city, as is hinted by the book in the Midwich Elementary library. It's arguable that this is due to some untapped psychic potential of her own, or that she was able to somehow tap into the power of the god that was incubating inside her.
Either way, Cheryl is potentially an extraordinarily powerful psychokinetic, capable of ripping holes in a city. She cannot deliberately access most of this power, and it's possible she might not have any of that power left after aborting the fetus of Samael. She can occasionally levitate a pencil on a good day.
She still retains a fair amount of control over "her" layer of the otherworld, however. Cheryl can fade into and out of her otherworld at will, and can take people with her (although she doesn't often do that, because who would want to go?).
Cheryl's also capable of arranging the "set design" of the otherworld. She can arrange for methods of transport, equipment, medical supplies, or food, to name a few things, to be available in the otherworld. If she arranges for transport, she can pretty much go anywhere she wants, like it's a very slow teleportation effect. It's how she suddenly appeared and disappeared in Liz's hospital room, for example. The exception is if she wants to go to Silent Hill itself. Usually, that takes almost no time at all.
She can sometimes use the link to the otherworld to learn things about people. It's more like being aware of what raw nerves the otherworld would tapdance on if she were to take you there, and rarely takes the form of anything specific.
Cheryl will usually know if your character has been to the otherworld, or if they've visited Silent Hill recently.