Cat 5e and Cat 6 are for the most part interchangeable. The cable itself is pretty much the same, it's the equipment (router, switch or network adapter) that will determine what speed it uses. A network adapter will always connect at the lowest speed possible. For example. If you have a gigabit nic (network adapter) on your PC and you attach it to a router that only supports 100mbps, then the link will be 100mbps.
The connection between your modem and your service provider is fixed by your service provider. Nothing you will increase that speed or incur any cost. Ethernet connection will only effect the speed at which your devices talk to each other and not the speed in which the access the world wide web.