If you're posting a lot of full-res photos, you may well run hit the limits of all those sites quite easily. From what I can remember, Picasa has a 1gb storage limit. Flickr limits the amount you can upload each month (100mb a month, I think?). Photobucket has a 500mb storage limit and a 10gb/month bandwidth limit. All three sites also have various paid accounts which raise or remove the restrictions on their free accounts.
Personally I have only used Picasaweb (Picasa is the software, Picasaweb is the online photo storage thingy) and quite liked it. I'm not all that keen on how it deals with displaying fullsize photos, but I like it more than flickr.
https://picasaweb.google.com/wikrok
As far as I'm concerned, Photobucket is the place Myspace teens upload their flashing GIFs and not good for much else

.
Now I use Gallery2 installed on my own webspace, but that has a monthly cost to it, unfortunately.
https://gallery.fuelledbycaffeine.org
I dunno if there are any decent free sites which will let you host lots of full res pics. If you can live within their limits, I don't think you can really go wrong with Picasaweb or Flickr.
Phil