The cat chased the rat.
The cat chased after the rat.
They walk across the road.
They walk over the road.
The dog jumped over the brick wall.
The dog jumped across the brick wall.
The birds are flying up/flying in/into/above the sky???
Either "the cat chased the rat" or "the cat chased after the rat" is correct, but they have slightly different meanings. "After" carries the connotation that the cat will not catch the rat. In a different context, "to chase after" a person may mean to pursue romantically (as Sanjay notes) -- but this is not its sole or even primary meaning (at least in American English).
"They walk across the road" is the more frequently used, but there are times when "they walk over the road" might also be used. Again, they have different meanings. "Across" emphasizes the crossing of the road from one side to the other. "Over" carries the meaning of walking above (say on a overpass bridge) or on top of the road.
"The dog jumped over the brick wall" is the typical usage, but again "the dog jumped across the brick wall" might be used to convey a different, but still correct, meaning. "Over" carries the connotation of going above while "across" suggests going from one side to the other.
Again, the word choice depends upon the meaning intended. Typically, birds fly in the sky. But when birds take wing, they fly into the sky or they fly up into the sky. If you want to emphasize that the birds are flying very high, you might say (poetically) that the birds are flying above the sky.
|link comment||edited Jun 30 '12 at 17:23 Jeff Pribyl Grammarly Fellow|