In the US, it's kind of random and doesn't necessarily signify anything. If the street is on a hill, up or down, could refer to the slope of the hill. In a larger geographic context, when traveling from state to state, up, down, or over, is sometimes used to signify cardinal directions on the map. Oftentimes, though, it can be somewhat arbitrary.

If the street is flat, you can say either, any time. But if there is a hill, say up for climbing the hill, down for descending.

Unless it is visibly uphill or downhill, I'd say "Follow this street" "Take this street" "Walk this street" (only if on foot) "Continue along this street" "Go along this street"

If you start at the bottom of the street, you go up it. And vice versa. Here in England it would usually depend on the house/building numbers