If you wish you can do this, as long as you set all your Gigabit equipment on a Gigabit switch it will work at gigabit speeds, but anything connecting over wireless or from a 100mbps only device will still operate incredibly slowly.
Also Cat5e will suffice for gigabit in most homes, STP would be more helpful. You just need to make sure the runs aren't very long, and that all wires are used in the cables, since Gigabit needs all 8 wires where as 100mbps only needed 4 to operate. (I still see a lot of "cat5e" cables being sold with ONLY TWO twisted pairs.)
BUT, if your only transferring at 2-3 megabytes (not megabits) per second, then something else is bogging you down, and it doesn't appear to be your network.