1) I am using a network card to try to connect my computer to an existing network running Windows 95/98/NT/Me, but I cannot see other computers even though I can see myself in Network Neighbourhood

Check the protocols that you are using. To run on a Windows network, the main protocol needed is NetBEUI which can be installed in the Network Neighbourhood properties.

2) I can see my network and communicate with other people on the network, but when I try to access the Internet I cannot seem to get on-line.

Within Network Properties, there is a protocol called TCP/IP (Transmission Control Protocol/Internet Protocol) which you will need to install to be able to browse the internet. Initially you will need to get details from your system administrator.

3) I have plugged my network card into my system, booted into Windows 95/98/2000/Me but the card is not being detected at all.

If you reboot your machine, go into the BIOS and reset configuration data (this varies between BIOSes but usually can be found in PCI/PnP Configuration) and then save and exit. Boot into Windows again where the card should now be detected. If it is not, then there is a hardware fault. Please see your vendor.