How to connect arduino with local network
To connect to a network you need a network interface.  This could be an Ethernet interface, a Wifi interface or something that supports TCP/IP.  The easiest way of doing this is with ether an Ethernet 'shield' of a WIFI 'shield'.  

In either case there are a range of options, from full shields to smaller boards that connect via jumper wires.  One important consideration is the longer the wires the more chance for signal degradation and loss of comms.  So direct plug in boards are probably better.  

Probably the easiest way to do it is to use a ESP8266-??, but using an ESP as a network interface is like only using your PC for email.  The ESP is faster and has more memory than the Arduino, so it would be better to program that as the web-server and get the Arduino to feed it the data.

You could use Bluetooth, radio, etc, but that's just making life hard for yourself.

You could also look at the Arduinos that have networking built in.