For sketching newsprint is the art school standard because it's cheap and it has a bit of tooth. Printer paper is very smooth, which makes it less suitable for drawing with graphite or charcoal. The tooth is what holds the graphite and charcoal in place.
A number 2 pencil is very hard, so you have to press firmly to make a darker mark. You should treat yourself to a range of graphite pencils; they are not very expensive. I use a 3B, a 5B, and an 8B most of the time.
I'm guessing that the eraser you're using is not a grey gum eraser. Get one. They are cheap, and they are worth it. You should knead it a bit before use to soften it. These can be shaped, so they are great for erasing weird tiny bits.
Because this is very common, I'm going to venture that, if not you, at least a few people who read this are artists who try to save time by turning a sketch into a finished drawing. No doubt a sketch can be very beautiful, but it's better to use your sketches as visual references for a finished drawing. If you buy semi-expensive paper, you may become worried about wasting it, and that can prevent you from sketching regularly. On the other hand, sketch pad paper doesn't look very good and is not archival.
Many people, myself included, don't even bother erasing everything in a sketch. I only erase something if it's visually confusing enough to keep me from finishing the sketch.
I often hear people bring up Caravaggio at this point. They will point out that he did not do drawings first, and preferred to work everything out on the canvas. For those reading who had this thought, I ask that you consider the difference in the mediums -- it is easy to paint over a mistake in oils, but graphite drawings are more delicate.
The best drawing paper for using an eraser would be something high quality, archival, and fairly thick. As I mentioned above, a graphite drawing is a fragile work of art. You don't want to be going nuts with your eraser directly on the expensive paper because too much will start to show up. But little bits here and there you can erase with a soft gum eraser, then gently sand with fine grit sandpaper.
Part of the beauty of a gallery-ready graphite drawing on paper is the illusion of there only being perfect marks on otherwise pristine paper. (Depending on your style, of course. Since you use #2 and mechanical pencils, and you want to use an eraser, it seems like a fair bet you're going for work with a clean and precise look.) And there you have the reason why artists will work it out on newsprint before hand -- unless you're some kind of a savant it's impossible to get this look in the first drawing.
Depending on the type of drawing you're doing, you might even want to experiment with cartooning a sketch onto archival paper. (Tape tracing paper to the drawing, trace it. Flip the tracing paper over to the other side, place it on a light-colored surface, and trace it again in a soft charcoal. Gently place the tracing paper in place on the archival paper, charcoal side down, and trace it again. The pressure will transfer the charcoal drawing onto the new surface.)
I hope this helped! Have fun drawing. :)