can you track someones activity with their pc serial no?

No.  The PCs serial number is not used in any way to identify a PC on a network or the Internet.

What a PC does have is a NIC (Network Interface Card.)  That is why does all the talking over a network or the Internet.  Each card as a base 16 number hard coded into the card.  It is called a MAC address.  That address is used to identify computers on the network, and it is linked to the IP address of the computer.  This MAC address is sometimes referred to as being "like" a serial number for the network card.  But not for the PC itself.

So no, you can not find a computer on a network using the computer's serial number.  

the only thing you can do with a serial number is to prove that you have it in front of you. this is why you should never give it to anyone. they may claim it as their own.

A computer does not transmit is serial number unless you have an anti-theft service like computrace but it doesn't transmit the actual serial number it transmits and encrypted serial number that corresponds to the software.  That information goes along with the IP address that you're using.

The common ordinary person does not have the technology to decrypt the software that anti-theft companies are using.  So you can't use that information to find out where somebody is at without the aid of law enforcement.  But if your computer has anti-theft software and it's stolen. The anti-theft company will be able to track down the location of the computer