Low Quality Raster Logo Reconstruction in Vector Format

Preparation 

I scanned the image in 600ppi with color mode of RGB (color) but you could do it in grey scale sense this is a grey, black and white image.

Clean Up in Photoshop

To clean up I first cropped the Bat Man Logo with my name. Then adjusted the color levels to clear up most of the gray pixels. Erased any left over grey pixels and used the paint tol to color the grey spot in the bat and rim. Saved the image as a jpeg.

Live Trace in Illustrator

I opened the image in Adobe Illustrator to see if it will smooth out the edges. But as you can see it did not with a live trace of Black&White. This was the only trace that had the best looking quality.

Logo Reconstruction Using the Pen Tool in Illustrator

In the end I used the point tool to trace the logo. It took two anchor points to make the rim at a 18pt stroke with a fill of yellow. I maintained the oval shape by holding down the shift key. I cut/pasted the oval in a layer to not interfere with bat shape (also turned the eye off and locked the layer). Named the Oval layer Oval to keep track of my layers. I then started working on the bat (in a new layer named bat). It took 15 points with a 1pt stroke and a fill of black.

File Formats

 I saved all the images in .Jpeg but for the last image above you could save it as a .gif only because I only have two colors. Its a smaller file but it still has good quality and that is what we need to maintain. 
Advice on Vector Creation
It was sort of difficult get the bottom of the bat to look symmetrical to other side. I would suggest to just look at the image and turn of the inner fill so you can see where to put your next point.