Image becomes noisy/grainy when I open it in Illustrator

Did you enlarge the image before bringing it into Illustrator? If so, you're probably seeing artifacts from enlarging something with such fine, thin, contrasty lines.

In my opinion, if you're doing what you said you're doing above, you're working backwards. It's really really hard to get a good enlargement of a raster line drawing. All the computer can do is add pixels and make educated guesses as to where those pixels go.

Instead, I'd recommend creating an Illustrator document at the dimensions you're eventually going to want. Then, use File>Place to place the image into that document (instead of opening the document directly into Illustrator). Then, run Auto Trace, and find the settings that work best. If you're enlarging the whole thing after that, you can just try resizing from there - or you may find the enlargement looks better after you hit "Extract". Either way, it's generally a lot cleaner to enlarge a vector image (the result of Image Trace), than it is to do so to a raster image (the .png).

It looks noisy/grainy because it is raster, made of pixels. Once you Image Trace and Expand it should look pretty smooth.