A flying broomstick is not simply a "normal" broomstick pressed into service as a mode of transportation. The tail, for example, is made of twigs, not broomstraw, which would make for a poor cleaning tool. The flying broomstick is a magical item with built-in charms.

Flying on a broomstick, particularly one which accelerates as quickly as the Firebolt, must necessarily involve some form of magic protective field holding rider to broom and shielding against wind and inertial forces. Without this, it seems unlikely that anyone could hang on when a Firebolt accelerates from zero to 150 mph in only ten seconds. It also seems likely that flying a broomstick doesn't involve actually sitting on the handle, which would be uncomfortable indeed. As a matter of fact, most illustrations we see in the books show the rider floating a little ways above the handle. The spell which creates this effect is called the Cushioning Charm, invented by Elliot Smethwyk in 1820.

Broomsticks are certainly charmed objects. They even seem to be at least semi-sentient. The best ones seem to respond to thoughts. When Harry was nearly thrown from his broom, Hagrid stated that a simple collision couldn't have caused it to act that way because "[c]an't nothing interfere with a broomstick except powerful Dark magic--no kid could do that to a Nimbus Two Thousand." This magic doesn't replace the skill of the flyer, of course, but rather protects and enables them to fly with amazing speed and dexterity. Even when the rider has fallen off the broom, it doesn't fall from the sky, but rather it drifts off on its own (SS , PA )

Quidditch broomsticks
Cleansweep 5 (CS7)
Cleansweep 7 (Ravenclaw team's brooms) (PA12)
Comet 260 (Cho Chang's broom, Draco's broom at home) (SS10,PA13)
Firebolt(the newest and fastest on the market) (PA4, GF8, 20)
Nimbus 2000 Racing Broom (SS5, 9, 10, CS4, PA1, PA9)
Nimbus 2001 (CS7) came out August 1992
Shooting Star (CS4, PA10) Ron's broom, old school brooms
Silver Arrow (Madam Hooch fondly remembers this) (PA13)

chronology of sport brooms (QA)
1820 invention of the Cushioning Charm
1879 Oakshaft 79
1901 Moontrimmer
unknown year - Silver Arrow
1926 Cleansweep One
1929 Comet 140
1934 Cleansweep Two
1935 first Atlantic broom crossing (Jocunda Sykes, on an Oakshaft 79)
1937 Cleansweep Three
1938 Comet 180
1940 Tinderblast
1952 Swiftstick
1955 Shooting Star
1967 Nimbus 1000
1990 Twigger 90
August 1992 Nimbus 2001
August 1993 Fire Bolt

The Firebolt
Released in the summer of 1993, the Firebolt is currently the fastest racing broom in the world. Harry saw a prototype in Quality Quidditch Supplies the summer it came out and was sorely tempted to empty his Gringotts vault to buy one (PA4). He resisted the temptation, however, and to his surprise received a Firebolt for Christmas from his godfather, Sirius Black (PA11). The Irish International Side flew Firebolts in the Quidditch World Cup (GF8).

• streamlined, superfine handle of ash, treated with a diamond-hard polish
• hand-numbered with its own registration number
• tail twigs of birch, individually selected and honed to aerodynamic perfection
• unsurpassable balance
• pinpoint precision
• acceleration of 150 mph in 10 seconds
• unbreakable Braking Charm
• when you pick it up then let go,  it hovers at exactly the right height to mount
• turns with the lightest touch, seems to obey thought rather than grip
• superbly smooth action

miscellaneous broom stuff

• Brooms are stored in a broomshed near the Quidditch pitch. 
• Hagrid, on a particularly cold morning, can be seen defrosting the brooms. 
• Filch uses non-magical brooms for cleaning, of course, and there are broom closets here and there in the castle. Peeves tries to put chewing gum in the lock of one and Harry and Ron lock Crabbe and Goyle into the one in the Entrance Hall. Harry and Hermione hide in that same closet when they've gone back three hours in time using the Time Turner, waiting for themselves to pass.
• You can buy a "broomstick servicing kit" which includes a Handbook of Do-It-Yourself Broom Care," a tin of High-Finish handle polish, and a clippers to trim crooked tail twigs. Hermione bought one for Harry for his 13th birthday.
• If you're in the market for a broom, the best source for information and pricing is Which Broomstick. There's an order form right in the back of the book. When Harry's Nimbus was destroyed by the Whomping Willow and his Firebolt was confiscated, he half-heartedly perused Which Broomstick looking for a replacement.
• besides the Quidditch team members, we see the following people ride broomsticks:
• Hermione in the Chamber of the Flying Keys
• Ron several time, including pick-up Quidditch games at the Burrow, trying out the Firebolt, and in the Chamber of the Flying Keys
• Snape when he referees the Quidditch match
• Neville (for a few moments, anyway)

• Braking Charm - the Firebolt comes with one of these (PA)
• Hurling Hex - Flitwick examined Harry's Firebolt to see if it might have a Hurling Hex on it. It didn't. (PA) This was possibly the spell Quirrell was casting on Harry's Nimbus to try to throw him off (SS
• Anti-Burgler Buzzer - since electrical devices don't work alongside magic, this must be a spell of some kind
• Cushioning Charm - invented by Elliot Smethwyk in 1820.  (QA)
• Horton-Keitch Braking Charm (patented, named after the founders of the Comet Trading Company, which makes the Comet Racing Broom) (QA)
• inbuilt Warning Whistle (spell included on the Twigger 90) (QA)
• Self-Straightening Brush (spell included on the Twigger 90) (QA)