Hook up Internet to second computer?

Yes if the router has a built in hub then you can use it to add more computers.

For security look at MAC filtering in the router options together with WEP and WPA

There is no need to use another router. Your FIOS modem has multiple ports to connect computers to. If you are hooking up a computer via a network cable you shouldn't need to enter any WEP key. Just plug it into one of the available ports. It should connect on its own. On the other hand..

If the computer is wireless internet capable then all you should have to do is right click on the Wireless Network icon in the lower right hand corner of your toolbar on the bottom of your screen. Choose "View Available Wireless Networks". Choose your internet connection. Click "Connect". It will ask for your WEP key. After you enter that you should be connected.

This is how it is done on my Windows XP based PC. I would need to know what operating system you are running to be more specific.

i think of the 8 port 10/one hundred quickly ethernet swap will paintings high-quality. it somewhat is a hub which you may wish the crossover cable for no longer a swap. Plug the two laptop's ethernet cables into the swap and then plug the modem ethernet cable into the swap (i'm assuming you have an ethernet modem and not a USB one). Assuming you're working DHCP on the modem that's going to right away hand an ip address and default gateway settings to the two laptop's and all would be high-quality. If this works thats all you have chose, no actual clarification why try to be a alternative router modem and so on... in case you're happy going throught the swap, then in case you ought to provide the swap back to paintings then have been given to laptop international and purchase a affordable swap (no longer a hub).

Yes, your router should be fine. The tutorial is in the router's manual. Be aware that WEP is insecure - use WPA where possible.