i understand that data send/receive among networked computers would be faster on a gigabit router but would there be a speed increase for internet connection?
would it make a difference for a program like skype to have a gigbit router?
No. Generally, a network is as fast as its slowest link, which -- unless you have an extraordinarily fast internet connection -- is going to be your broadband link, which is likely a theoretical maximum of 6-15 Mbps, nothing close to the (again, theoretical) maximum of 1000 Mbps on a gigabit network. As you note: data transfers on your LAN will benefit tremendously. Skype, etc. will not.
Actually, this is true for all data paths in computing, which is why it's really useful to memorize the important parts of a bandwidth chart like this when you're starting to performance tune something.