Modern Vespa

There's some good advice here.

If you're a new rider, you're probably worried your tires will slip if you take the corner too fast. On a dry, debris-free road they won't - you'll drag the centerstand or "security loop" first. You can crank up the preload on the rear shocks to minimize this to some extent, but know that dragging hard parts is never a good idea. (You can also hang off the scoot to make it more upright through a corner, but this is not a novice rider skill.)

The suggestion on Proficient Motorcycling is excellent. In particular, Hough's description of the optimum "line" through a corner is, for street riding, very good. As for how much of your lane you use, remember that you don't want your head to go over the centerline, but you also don't want to be leaned over right at the rightmost edge (in the US anyway) of the road. A lot of gravel accumulates out there, and when you hit that, you will lose traction.

The other thing is that many people, when they are taking the corner "fast" tend to charge into the corner too hot, and then have to either a) lean farther than they're comfortable with or b) try and scrub off some speed mid corner. Generally, unless you're a racer, b) is a bad idea, because you'll either lose traction and crash, or stand the scoot up and run off the road and crash. So it's a better idea, and safer, to enter the corner at a speed you're comfortable with, and then roll the throttle on to get more speed on the exit.

I like the suggestion of doing the same route over and over - it's one of the advantages of doing a track day, you get to repeat your actions at successively higher speeds.

Lastly, have fun and don't stress. If you find yourself scared while riding, back off and don't push it. It'll come with time.

I can second that suggestion. I learned by taking the same corner (both directions) over and over again until I got comfortable. It was a long, oddly designed curve, not well engineered, so it proved a real challenge--both entering and pulling out. After about 3 or 4 dozen attempts over the course of several months, I began to feel reasonably proficient--and, most importantly, other better designed corners were suddenly a breeze.

When playing with different lines through a turn it's better to do it way slower than "the limit".

You need to slow things down to allow time to make every movement a conscious act and to observe the results. Every time I'm stuck behind a slower moving vehicle I move into practice mode. If I enter the turn here what can I see? how hard am I leaning when I apex? what does my trajectory out of the turn look like? how does that set me up for the next turn?...

I also do this whenever sweeping on a ride but there I add observations about the other rider. How do our lines differ? how does that change our relative positions?...