I just went back and read the section about the bewitched broomstick in P/SS... the only wands mentioned anywhere near there is that Hermione uses her wand to set Snape's robes on fire.
So some spells seem to require wands, some don't. Some spells can be done without wands, but wands make it easier. Some spells have to have words uttered, some don't, some are so fickle they depend on you saying the words correctly and using the correct flick of the wrist (ie. Ron trying to do wingardium leviosa in P/SS)... some require absolute concentration, others don't. This must by why wizards have to go to school, they just can't learn every little thing from books, because their power has to develop, and their knowledge of what to do to cast and direct spells correctly has to grow...