If I want to use CS3 on a second computer when I'm in another city, can I simply install it or do I need to uninstall it on the first one before installing on the second one and back and forth?
You can have the software authorized on two computers at any one time, and they don't have to remain on the same two computers. Even after you have authorized two computers, you can deauthorize CS3 on one or both and then authorize it on another computer. So don't forget to deauthorize on computers on which you are going to stop using the software, such as when you sell a computer.
There should be a "deactivate" or "deauthoize" item in the "Help" menu that will bring up a dialogue box. I don't know which version started using this activation scheme--may or may not be in 1.5.