How to generate wallet address? And how to get that connected with internet for transactions? I would like to know how a wallet address can be generated? One site suggested me to create a script for that, that is 1 option, I would like if any other there.

Mainly I am willing to know is how that address will get connected to network for transactions?

If I have my wallet address and I give it to anyone, will the other be able to deposit bitcoins in that address, using my wallet address only? If yes how will the network come to know about my wallet address?
You dont have to be connected to the internet to have your wallet verified or something. People can send bitcoin to you (or a random valid address) even without them knowing it exist. You can create an address on a site like this or create an address in your wallet app (such as Multibit, which is a program I'd advise to use). I would suggest the latter option, because it is safer (just 1 site/app that could possible know your secret key) and it takes away the hassle of importing the wallet in Multibit. 

Your second question: yes, the address is the only thing the other person needs to know. Sometimes you also have a QR-code which act as the same thing, but your address alone is enough. 

TL;DR. No need to connect to the internet, and create (and control) an address in Multibit. The one who is going to send you coins only needs to know your address. Good luck!