Should I just quit riding after only a week?

Nicole I understand that you do not need a very fast bike. One other reason you should get something bigger other than a 250 is that they bog down going up hills so you would constantly be down and up shifting. It becomes a pain in the ass big time. Remember if you get a 600 it only goes as fast as you want it to go. I really recomend you consider a 600 but its your bread thats gettin spend and you have to ride the bike.  Congrats on your license!

                                           Mantis  

SoutBayCruiser/Fast in Pink
If you wanted to ride in the first place, a good scare will not cause you to park the Bike, and should instead show you what not to do!  Being a Member of the 'Over the Handlebars Club', I can tell you that its no fun stacking your Bike, but if you really want to ride, you will get back on with a greater understanding, and respect for your Steed.
Motorcyces are a Physics/Odds Game.  Understanding this is key to being safe, having a good time, and surviving each and every ride.  
The Physics aspects don't require a Degree to understand.  It is Gravity vs. Enertia vs. You at all times.  The Bike wants to stand up under thrust, and fall over in decelleration, so easy off the throttle into the corner, while adjusting your body wieght to lean into the Gravity, and hit the thruster coming out.  Easy stuff once you find it.
The 'Odds' are always against you, whether or not you are new to riding, or have 150 years of experience.  Every ride is different, even if on the same route each day.  Autos, Road Debris, Weather Conditions, and even your Bike, can effect how that corner feels each time you take it.  Every Second holds a constantly changing situation.  It really does take full concentration, and focus, to survive.  The more you train yourself (can't be learned in a class) to focus on your Bike, how it is responding, and your surroundings, the easier it becomes second nature.
As for the Older, Inexpensive Bikes, the same laws apply.  Physics Laws state that everything wears out. Odds are, even that pretty Bike with low miles will likely need some kind of work.  The best deal might look good at first glance, cause the Owner bought it new, scared himself to near death on it, and parked it in his garage for the next 5 years!  The Oil now has high Acid Content(bearings), the Tires are Hard (lack grip), and unless he drained the Fuel it will need a serious cleaning!  Expect to do some work to it.
Personally, I prefer the Older Bikes, and enjoy working on them.  I ride a 1986 Kawi Ninja 1000R, that I trust completely with my life, as I have torn it down to bare frame, and replaced all the parts I didn't like!  But this kind of diligence is not for everyone, and it is a nice feeling to have a Newer Bike, and not have to stress on if it will take the next corner, allowing you to focus on your ability to ride.  However, either New, or Old, All Bikes Need Constant Inspection/Maintenance.  If you treat it like a Car(get in and go), it will disappoint you, and may even kill you.
Start Small, and work your way towards the Super Sport Bike that you really want, and first saw yourself riding(it will still be there when you are ready for it).  Expect to have a few scares along the way, and sometimes everyday!  And most importantly, understand that nothing about riding a Bike(any Bike) is 'forgiving'.  It is up to you, your ability, and constant focus, that will keep you alive, and enjoying the ride!  It is an extremely demanding Sport, both Phyically, and Mentally, but it is Equally Rewarding, and there is nothing else like it!  Ride Safe, Be Smart, Stay Focused, Wear Gear, Trust No Motorists, and Always Expect The Unexpected!  

I wouldn't quit riding, I'd just ride within my skill level. Take it easy and read about riding tips in Sport Rider, you can actually learn alot. The best thing that will help though is experience, try to ride every chance you get and ride within your limits, you'll get better. I have been riding for several years and have owned and rode alot of different bikes, sportbikes to me are the best. They handle better, have better brakes, and they are lighter also. Don't quit, nothing is better than being out on the open road and enjoying a good twisty ride. take it easy until your skill level gets better.  

Yeah, go racing on the street where there is gravel and oil everywhere. Old geezers pull out of their driveways right in front of you, cars change lanes and cut yer azz off. You'll be spending a lot more for new plastic and hospital bills than you would ever spend at the track. Wanna-be racers are just squids, dum-fuk.  

A man has got to know his limitations.  You will never regret obeying all traffic laws.  Keep your speed within the posted limits and come to complete stops.  Wear the gear.  Ride well rested.  And never ever mix the bottle with the throttle.  Go to the track to see how fast you can drive.  We have all had high pucker factor experiences.  Your constant vigilance is your best protection. Quit riding?  Never!  

Nicole - congrats on the license, you'll have a good time and save some dough on gas, that's for sure.  After having started street bikes on a 750 and never riding anything smaller... got an old 99cc kawasaki for the kids and wife to learn on last year.  Well... it tops out at 55mph, but I have to say, in some basic ways it is more fun than anything I've ridden before.  I'm considering buying a the newest 250 ninja, myself.

As for the o.p. - don't be in such a hurry.  

Dont quit, just take it easy. My first ride was a Honda 250. Next a 400, then bought a KZ1000. WOW, that was a rush.  I rode that bike for 3 years, 365 ! Recently joined the Over The Bars club as a result of an ill timed u-turn by a car!  Totaled my ZX-10, but my new ZX-14 is on the way!! The point is: you ride at your pace, dont let others dictate for you, and allways HAVE FUN!!  

I would have to say "don't quit!!!" riding is so much fun, its freedom, its the air in your face and your tires on the ground!!! i lost my virginity 08/20/07 and i think i would have to say its better than sex, maybe im not doing it right though?  but the freedom of ridding has its negatives, your unprotected and people just don't see you, adding any mistakes to that and it can be unforgiving.
i started out on a 1991 suzuki 750 katana, its a beast! and the butt puckering experience happend a week later!!! i took a 175 mile trip (all highway). of the first 60 miles the last 40 was a never ending butt pucker.  i didn't want to go over 70. it didn't feel right!!! crosswinds, big riggs and sports cars flying passed me pushed me left and right, other bikers laughing at me, but it was me riding at my level and not there's!!!
it took me a half hour and tossing up a freshly eatten Mc Burrito to get back on the bike. i even thought about turning around and heading back home! but after that it was 100 miles of easy going, every one driving about 60 and enjoying the scenery!
after all that i got too confedent in my 'abilities' and tryed to keep up with my Dad in his truck, i took a turn a little faster than i should and my bike wasn't responding. i ended up in a ditch and joined the Over The Bars Club after a week of riding! but guess what? i got right back on and rode the rest of the trip up and the 175 miles back a couple days later!!!
i still work with getting better with my riding abilities to this day! i had a hell of a fright with down hill curves!!!  scared the living *** out of me and would send me into a panic and i would slow way down. now they just make me pay a little more attention to how the bike feels and how i'm sticking to the road.
so don't fret about your brush!
Feel The Fear, Respect The Fear, Love The Fear!!!
Feel The Ride, Respect The Ride, Love The Ride!!!  

Good for you "Fast-In-Pink"!! You're doing it right! The 250 Ninja is a great bike...Honda made a 250 sport bike for a while the vtx 250...You go girl and stay safe!
P.s. My wife rides a Honda 250 Rebel...  

No matter what you ride, never ride faster than your angel can fly.  

Well its only a 600, blah, blah, blah.  This bike is faster than your Rousche Mustang, and requires alot more skill to operate.  You might trade it in on a less aggressive bike or do a trackday, but definitely do a rider safety course, and read a good motorcycle skills book.  I recommend Sport Riding Techniques.  You should also get some experience on a dirt bike.  Also come to the realization that it takes time to develop your skills, and take it easy, you are not Rossi, yet...  

Don't blame the bike for you being stupid.  It really doesn't matter what you ride, if you ride beyond either your riding skills or the bikes handling, you can hurt or kill yourself.  Slow down, ride within your capability, and don't let you ego or adrenalin cause you to do something stupid again.  It really takes years to be able to judge and react to most things you will face.  If you can wait or cannot control yourself, get rid of the bike before it gets rid of you!!