Well as the owner of a "fairly big website" you are grossly misinformed. The average .png image is SMALLER than the corresponding .gif image. Where you got your information from, I don't know, but when I convert png's to gif the filesize increases. When I convert gif to png the filesize decreases, similar results are reported by numerous online image groups.
"[PNG is] perhaps about 25% smaller than TIF LZW for 24 bit files, and perhaps about 10% to 30% smaller than GIF files for indexed data. "
burnallgifs.org references a letter written by Christopher Wright (an online-cartoonist) who states:
"Switching over to the PNG format has given me about a 4K reduction in size per panel. "
Additionally, burnallgifs.org references a post by Drake Emko (another web cartoonist) that states:
"PNG images compress more efficiently than GIFs. The smaller file size you can achieve with PNG may be the most important reason to switch to this format."
Want some hard evidence rather than opinions?http://www.algonet.se/~otsu/pngcomparison.html
Note the size of the compared checkerboard image. .gif was at 0.99kb, where as .png was at 197bytes. Thats a signifigant difference.
Ok, now you might be saying "Well these tests were conducted by just random people! That doesn't mean anything!"
So lets look at some test results from Yale University (http://www.library.yale.edu/wsg/docs/image_pro_con/imgprocon.htm
"[PNG] typically compresses images 5-25% better than GIF."http://www.oit.umass.edu/publications/at_oit/Archive/spring00/jv_compress.html
also contains a very in-depth and accurate description of jpeg, gif and png, and it also states that png compresses files smaller.
So if you want to say .gif is smaller than .png, thats fine, but all the evidence suggests otherwise.
Oh and to rogue, who mentioned about IE's lack of alpha transparency support. Yes you are right, IE does not support this, however neither does gif. So why does that make png useless? It's not like if you use gif you can use alpha transparency, so your argument doesn't make any sense to me at all. It's like saying you have a choice between A and B, B has signifigant advantages over A. However, both A and B lack a feature. Therefore you should use A. Well A doesn't have the feature either, so all you're doing is losing out on all the added benefits that B would provide. That makes no sense at all...