The resolution of photographs you post online to Picasa depends on why you want to do that. Do you share photographs with friends, let everyone download a portion of common pics or want to use it as a backup?
I do not use Picasa, but I use Panoramio, also by Google. When I upload photos that I know I will reference in my blog, I use 900x600 resolution, because this is the one which suits my blog's design the best. It also looks good when viewed on Google Maps.
The decision also depends on the circulation breadth of your photographs. If you want people to use them as wallpapers (even without your consent) or print them out or use in some third-party application like private invitations or pocket calendars, then a typical HD-screen resolution would suit you fine. I do not enjoy the thought of losing track of my photos on the Internet, that is why I prefer to scale them down to 900x600 px. This is a good resolution for viewing images on most standard laptops, and it will look no good if printed out.
Another possibility, as D3C4FF mentioned, is that you plan to use Picasa as a back-up storage. In this case, then upload photographs in the highest resolution possible. I do not know about you, but personally I do not trust online storage resources as much as I believe in the power of hard drives. If you want to back-up your photo collection online, you might consider creating an offline storage for that.
Forgot to comment on the question of image quality. I do not know what other options are available besides "automatic", but in Photoshop JPEG quality level 9, in Lightroom quality 80% are usually more than enough.