With an iPhone, is there a way to automatically sync photos taken with the camera to Picasa
You can sync using iTunes' "Photo" tab, that would be two-way but isn't Picasa like you asked for. But if you set Picasa to scan that folder, then you'd have it. 

Otherwise you could download photos from your phone to Picasa, but it's one-way only:


Connect your phone to the computer.
In Windows, click Start > Settings > Control Panel > Scanners and Cameras.
Right-click on "Apple iPhone" > Properties.
Click the tab "Events".
Click the radio button "Save pictures to this folder", and select a destination that Picasa scans. Then OK.