First, let me say that I have thought about this a lot, given your comments about using your neighbor's internet conenction. By doing so, he is probably in violation of the terms of use agreement with his ISP. So telling you how to do this would be unethical.
That being said, I can offer some general comments.
What we now know, is that you are using a wireless setup. In order to get wireless going you need to know what the chipset is on the wireless card that you have. Many manufacturers do not support Linux, so you need to use/find drivers for Linux, depending on your wireless chipset. Some support WPA outright, some do not. End of comments on this.
You should use the wiki for Puppy to learn about using Linux, since that is the distro that you are using. Different distros have different support for hardware and ways of doing things.
The saving settings issue depends on how you are using Puppy. The Live CD or a hard drive installation? If the former, did you save the settings on exit, as prompted, to your hard drive, or omit this process? Saving them on the screen using the Live CD of puppy is one thing which saves the settings for that session. Saving the session to the hard drive will allow you to use the Live CD again with the settings manitained, as Puppy looks for the saved setting file on the hard drive on your next boot.