Is there anything i can do to a room to prevent as much sound from escaping to the outside or the room next door at a very low cost?

First, seal all the gap on four side of the door using wheather strip available at Home Fix. Double the strip if necessary. Should cost <$10 per door. Next, if your door is hollow, change to the solid one. 

your windows need to seal off too, double glaze windows the most effective.