The gigabit ethernet port on your computer is also backwards compatible with 100 Base-T. So yes, you can use that, but it will only run at speeds up to 100Mb/s.
I'm a little confused with your question. If you want to add gigabit to your network and still access the internet, You don't need a gigabit router. All you do is add a gigabit switch to your existing network. This is done by connecting a Cat 5e or 6 (reccommended) cable to the existing router to the Gigabit switch. This will provide all your PCs access to the internet, provided you use one of the other ports for the Cable Modem. All other computers on your network should now be connected to the gigabit switich. Everything should work as it normally does.
Here is a quick pic: