When you scan in something that’s been drawn on paper and enlarge it, you’ll see the irregularities of hand-drawn art. They’re not a bad thing if that’s what you want, but if you want smooth, tight, even, vector artwork, you can’t really start out with an analog drawing that doesn’t have those qualities to begin with.
If you are intent on this, though, use a hard-surface, non-absorbent coated paper and draw with India ink or some other kind of ink that won’t leave a rough edge between it and the paper. This will reduce the irregularities of where the ink meets with and soaks into the paper.
As for your scan opening in Photoshop, that’s because it’s raster (bitmapped) artwork. If you want to convert it to vector art, you’ll need to do so in a separate process using something like Adobe Illustrator. That is unless you have a scanner (like I’ve never encountered) that will scan analog art and automatically convert it to vector art.
As for the process of converting raster art to vector. it usually doesn’t process satisfactory results. Vector artwork is composed of lines and fills and the points that define them. It just doesn’t lend itself to more complex artwork, like painterly effects or irregular edges. The results always look awkward — especially when enlarged.
And just to be a bit pedantic about things, there’s no such thing as a dpi in a raster file. Dots Per Inch (DPI) describes the number of halftone dots on a printed page (or their toner or inkjet equivalents). Pixels Per Inch (PPI) is the correct term for, well, the number of pixels per inch in a Photoshop (or any raster) file. They’re very different things.