One important difference from shooting color is that scenes that look interesting in color may not be so in black and white (or the other way around, as well). We are used to judging scenes by color contrasts, whereas the BW film is only going to see shades of grey. Try to look for interplay of light and shadows - makes much more interesting BW photos.

Look for interesting textures (e.g., weathered wood, rocks).

Also look for interesting patterns (e.g., marks on the sand at a beach showing ripples due to waves - may look dull in color, but can be very nice in BW).