It does send it through directly at full speed and does not bother the router. That's the whole idea of a switch. Moreover, if Port 1 and 2 exchange data, port 3 and 4 can do it at full speed whereas port 5-8 connect to the internet etc.
I believe (I'm no networking expert)that an ARP request is sent and the switch then learns the MAC of the connected network device and the port it's connected to.
All traffic that device is then sent to the device on the switch.
Only thing the router might be hit with is some broadcast traffic and ARP requests to find out what's where.
A switch unlike a Hub (Hub will broadcast everything to all ports) has intellegence and knows which port a device is connected to even if there are other switches between it and the device.
Even the el-cheapo unmanaged switches can remember at least a couple hundred if not a couple thousend mac addresses.