png's or gifs??

If you use only one colour background, no. Otherwise, yes.

Only png can deal with transparency (32bits, with alpha channels), so if you want to use your graphic elements with different background colours (light or dark) AND want those graphic elements have smooth borders, choose png format.

The inconvenient of png format is its size. You can use this nice software to optimize your png files: PngOptimizer
 
Is there a problem using gif's instead of png files for things like smilies and other icon type images??

It all depends. If the images are static you might as well use pngs as they offer transparency which gives don't so you know if you have multiple styles Example: dark bg style and a light bg style .pngs will look good on those styles without any matting issues occurring. If the images are animated you won't have an option but to save as .gif since png doesn't support animation. Drawback is that if you have multiple styles (example i mentioned) you have the issue of running into matting issues.

If the image is static - png
If the image is animated - you have no choice but to use .gif
 
If you use only one colour background, no. Otherwise, yes.

Only png can deal with transparency (32bits, with alpha channels), so if you want to use your graphic elements with different background colours (light or dark) AND want those graphic elements have smooth borders, choose png format.

The inconvenient of png format is its size. You can use this nice software to optimize your png files: PngOptimizer
or just save the png file with adobe...it has an option to "save to web"
 
or just save the png file with adobe...it has an option to "save to web"

photoshop doesn't optimise them as good as some optimisers. I know I can retain quality and lower the file size significantly with fireworks (but that's paid and overkill to pay for something to save stuff) so for png-8 just about anything is better than photoshop if png-8 and optimisation is the OPs cup of tea.
 
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