When or how does one becomes a part of Bitcoin P2P network I am new to bitcoins and got a question raised in my mind recently. I learnt that every person doing a transaction is a node in btc p2p network and that when he/she joins the network he/she gets a full blockchain starting from the genesis block. But can anyone tell me that in reality am I becoming a node while I am registering myself on a trading site like Zebpay, Koinex or IQ Option. If so then how can I see the blockchain I have received.
When you sign up for a wallet or exchange site, you are not becoming a node on the network.  You are using a site that already runs several nodes on the network, and you are trusting them to send and receive transactions for you.

To actually join the network, you need to run your own node.  This can be a full node or an SPV node, but the point is you are running it yourself.  Otherwise, you are using a 3rd party to connect to the bitcoin network on your behalf.