1. PNG is great, but I can't use them because they don't work in Internet Explorer.
This is one is almost entirely false. PNGs of any flavor will display in Internet Explorer, even back to some pretty old versions. The problem here is the alpha transparency of PNG-24's doesn't work in IE 6. That's it. if you don't need alpha transparency, you can do anything with PNG in IE that you could in any other browser. If you DO need alpha transparency, even that isn't too big of a deal as there are lots of ways to deal with it.
2. PNG has the smallest file sizes.
When comparing GIF to PNG-8, PNG-8 often has better compression. Not always, but often. But if you are talking about PNG-24 (The ones with alpha transparency), they never have the smallest file sizes. In fact, they are often nasty large and huge culprits for slow loading pages.
3. PNG has better color
Again, this is an attribute of PNG-24 only. PNG-24's have great color. PNG-24 even has lossless compression, meaning you can save it over and over and never lose quality! But of course it comes at the expense of file size. PNG-8's use the same color-indexing and have the same color limitations GIF does.
4. PNG is the all-around best format.
Completely false. Each of the three major image formats, JPG, GIF and PNG, all still have their place as the best format for the job. For photographic images JPG still cannot be beat in terms of file size. PNG-24 can replicate the quality, but at several times the file size. PNG-8 beats GIF almost all the time, but PNG can't do animation at all, so any time that is needed GIF is your format.