I would say that Photobucket is the main site that I've seen people using. I've used it for over three years now and only occasionally ever had a glitch with it--certainly no more, or less, than any other online site or service. The only other photo sharing site that I tried and can report nothing but disappointment and frustration with was "Webshots"--stay away from that nightmare, IMO. High speed internet certainly helps, as does taking a few minutes to actually read the "How to use" section of the site. When all else fails, read the directions!
One of the most important issue affecting photo quality is lighting. If you're hoping to get a good photo with just one single, standard 60-watt light bulb, you're going to be disappointed nearly every time. Your photos will have a yellow tint, and harsh shadows. Get either fluorescent bulbs, or shoot in natural light--I have a huge picture window that I use to light my shots, augmented by a fluorescent overhead.
As far as taking photos for the website, first of all, make sure your camera is at least decently new; that is, one preferably less than two years old. Another trick is to use a photo-editing program like "PaInt" to "hone in" on your photo subject, and use this feature to "zoom in", rather than trying to get too close with the lense or camera zoom itself. Many times, the zoom feature can give you problems in reducing depth-of-field, and you get sharp focus in the foreground, and then blur along the length of the model. Shooting from further back, and then "zooming in" by shrinking and centering the photo in "Paint" can be a huge boon.
I also use Microsoft Office Picture Manager to correct tones and lighting. Open a photo with that program, and select "Auto Correct" and watch the magic, In many cases, you can work miracles with this convenient program.
Hope this helps you improve!