lucky you, receiving a bouquet of flowers!
and yes, you can do the sugar thing (not too much) or you can add seven-up or sprite to the water (it's the sugar in that that is good), or an aspirin ...although i don't why...
but the best thing you can do is just put a 1/4 teaspoon of bleach in atleast one litre of water.
it discourages the organisms that can clog the stems and make the flower stalk unable to take up water... thus shortening the life of the flower considerably.
i come from a european family that bought flowers every week with their groceries. flowers were as important as buying milk and bread. i still do that and i still do the bleach thing. i do it because it works! i've experimented and there's no doubt that it does lengthen the amount of time the cut flowers stay fresh and turgid and beautiful.
and yes, definitely cut the stems again, and every two or three days thereafter when you refresh the water. depending on what kind of flowers, you should cut the stems on an extreme angle (diagonally - so it has the greatest water uptake surface) and do so under water or under the tap if possible. this makes sure that the stem does not dry up at all and subsequently block the water uptake capabilities.
if you really want to be anal about it, you can put the flowers somewhere cool at night or when you're not home. the warmth of the sun and home also shortens the enjoyment span of the flowers.
long answer, but these are the tips that i think are invaluable.
enjoy your flowers!
just always have flowers in the house.
always have.... always will.