Here's another idea. I think brooms should be more companion-like, instead of just holding them in your hand. Right-clicking your broom on the ground will cause it to spawn there, and follow you around like a pet. Of course you can right-click them again to put them back in your inventory for classes. Each broom has a different particle trail, to contribute to broom lore. http://imgur.com/f2YcuZQ
How you would fly, is by saying "Up" like in the Harry Potter movies. Sometimes it works, sometimes it doesn't, depending on your broom experience. If it is successful, your broom will fly over to you and you can then control it like normal. When you land, it would deposit you and continue to follow you unless you put it away.