Compared to what you have been used to, Safari is secure. All browsers, including Firefox have their problems but Safari and Firefox are closely monitored and patched very quickly, if necessary.
Glad to see you found the Mac Forum, you will be able to get better answers to your Mac questions than on the other forum.
In my opinion, there are no major problems with using Safari or Firefox. It's whatever you feel comfortable with.
"The mouse response is slooooooooooooow and sort of wavy, like it is drunk. We bought an Apple mouse, which is quick but way too stiff for my arthritic hands. Can I do anything which will make that mouse run tightly?"
Yes, go to the System Preferences, in the dock, and then to Keyboard and Mouse. Select the mouse tab and adjust the tracking speed, etc.
What you should have received in the box, along with everything else, was a VGA display adapter. This little dohickey plugs into a port on the back of the iBook and you plug a regular monitor into the other end of it. Check the box again, it was included with all versions of the iBook.
Check back on these three Mac Forums for further updates.
Let us know how you are getting on
Well done indeed on the Wireless network.