Does anyone know if an illegal in the United States can travel by greyhound within the united states without a problem, or is it like the airport where they check your visa?

An illegal can travel by bus without checking visas. Only airports do that. Just be sure you tell them that you are an illegal so they can call the cops on you.

Do you mean traveling from Canada or mexico and travel into the US by bus, or do you mean only traveling within the US.

Illegals travel by bus within the US all the time, and I would guess they travel by plane within the US, ( no international)

They clearly said "within" the US Chuck.