Many times a well ridden 250 can hang around the bigger bikes on twisty tracks. Their lower weight results in a far more agile bike that you can throw through the turns.
Don't be in a hurry to go bigger. Sure, they are quicker on the straights and have a little more acceleration, but you'll take a hammer to the people who think a bigger number means a better bike once you learn to truly ride that 250.
I'd suggest taking it to a track day and talking with the tire distributor. A couple laps with them watching will give you a worlds more experience and FAR more valid suggestions than asking here ever will.
JJ650's answer is somewhat right and somewhat wrong. Tire compound is extremely important in turning. However, softer compound does not mean the tire can support less. The amount of weight a tire can support is limited solely to it's air pressure rating.
On an ideal non-deforming surface, side-g force can only equal the frictional force created by the coefficient of friction of a material and the normal force to the road. However, tires are far from ideal and contact area plays a part - more contact area on a tire means more traction down to the road because the two surfaces interact in a non-ideal way. You must balance your tire pressure with the ride and suspension however: this isn't a "let air out of your tires to improve performance" sort of thing.
If you ever take your bike to a track day, the tire distributor I mentioned above can walk you through what you should be looking at: and it varies depending on your riding, the track you're at, etc. You will never want to use the same compound the day racers use though: that's far too soft and you'll eat through them at an alarming rate riding on the street every day.