It's good you're starting with hands. They're a good thing to learn early, I think.
For better accuracy when drawing hands (or anything, for that matter), try blocking them in on your paper very lightly so you can see the form before you start laying down dark lines. Also, if a line on your paper doesn't quite match what you're seeing, don't erase, don't go back over it (this can become a very bad habit), but keep moving, then correct the mistake next time (assuming you're drawing multiple hands per page or per session; for pieces you intend to showcase, it's best to erase the mistakes early in the light blocking-in stage).
Eventually, you're going to want to shade them in with the side of your pencil rather than making lines with the tip, but you can probably hold off on that until you're more comfortable with the contours if you want. If you don't mind trying it out, just go for it. It might look bad, but that doesn't matter. It's practice. This should theoretically help you understand the forms better -- it's a painterly way to do things, but it works well in less messy mediums too.
On the inked picture, yeah, ink pens take a while to get used to. I'm not good with them, but the theory is that you're supposed to draw smooth, confident lines, quickly. That way you don't get wobbly, jittery lines.
You did better than you think, though. You drew each line cleanly without any petting (except for some bits of the hair, which I'll get to later), and that's commendable. On the downside, as you've noticed yourself, her body isn't very convincing (though I'd argue the clothes work well at the angle you drew them at, even if they're somewhat vague). Anime characters (girls, at least) tend to have very thin bodies compared to their heads, but rarely as small as that girl's. I like to use the proportions of a real-life child around nine to 12 for traditional anime girls -- their shoulders stick out past the width of their head, but not a lot. Like so:
That's not set in stone for anime style, of course, but when in doubt, lightly draw a box to indicate the width of the shoulders first.
Onto the eyes. When you close your eyes, your lower lid doesn't move much. Your upper lid closes down over your eyeball. So...
In reality and in most art styles, the top picture is how it works. The bottom picture is how you've got it currently. It's an easy fix, though, and you generally only have to see it once before you get it down.
All right, hair now. I found it kinda odd that your lines were smooth everywhere but the hair, where lines are generally the smoothest. Draw hair with quick flicks of your pen to preserve the wispy feel of it. It's harder to get precise lines like that, but hair isn't very precise anyway. (I usually go in with a light pencil and block everything in first. Seeing a pattern here? Lots of light blocking really helps.)
As far as the rest of the face goes, try and keep the mouth placed where you could feasibly stick a jawbone and some teeth beneath it. This being anime style, what you have is actually fine given the style's distortions, but I think it's best to get into the habit of putting things in the right places most of the time.
After all that, here's a disclaimer: if what I say ever conflicts with something your friend's art teacher says, listen to the person who teaches art for a living. Ultimately, I know very little about art compared to someone who's gone through college to become an artist.