Portable Network Graphics is a non-lossy bitmapped image format, created to improve upon and replace the GIF format (copyrighted by Compuserve). PNG supports palette-based (palettes of 24-bit RGB colors) or greyscale or RGB images. PNG was designed for transferring images on the Internet, not professional graphics, and so does not support other color spaces (such as CMYK). It is unfit to represent quality photos.

TIFF and BMP is better, especially with (non lossy) compression like ZIP. PNG has a "transparent" color but what is really needed is a format that contains an alpha channel (transparency) so we could get rid of the overwhites and the clipping paths for isolations.

If you need to print  for display, for example, which would be the best format to save your work?
and at what resolution?  for 8by10 or 8.5by 11

Print is a totally different question. It depends on how they print it, by what technology. The last quarter of 2008 I did some fashion work and they needed posters for shops in malls. It was printed on a giant inkjet and they wanted the TIFF, but no noise reduction whatsoever.

For some reason the natural sensor noise looks much better on print than on a screen. Maybe it has to do with the paper or inkjet grain taking over in smooth gradients. Colors and luminance looks different on a screen than on paper. A printerguy will take your TIFF and then tweak the image to look best on a poster. It's their job...

In general, for print your image should look too light on a screen.