Of course you can show your picasa images in the forum. But I think google prefers to allow only the resized thumbnails to be displayed in forums, with the image linking to the page where it is supposed to be viewed in picasa web albums. There you can view bigger versions if you so prefer and also any EXIF, if available. I tried to display the original large pics too, by downloading them and getting the url from there. But it appears that google is smart enough to detect and tackle that trick. So, if the intension is to show the uber large pics with all EXIF, then picasa may not be the way to go. There are lots of free hosts like imageshack.us, radikal.ru, photobucket, flickr... ... But if you can live with uploading comparatively smaller ones which link to the bigger ones, then picasa offers a really addictive set of features including but not limited to organizing your pics offline and online, quick-retouching the pics before upload, and lightening fast uploading (with automatic resizing or full size - whichever you prefer). So, it depends on exactly what you want. I personally have accounts in picasa, flickr and photobucket, with flickr proving to be the most weird. When I want to show large pics with EXIF, I use photobucket. In every other occation I worship picasa.
Here's an example of displaying images here and linking to bigger version in picasa. The picture is clickable, and will take you to the address where you can verify if this really is a crop from a 200 ISO non-macro shot with one point-and-shoot SX10
This is what the above code does: