It isn't necessary have the bitcoin COre to start to use Bitcoin. If you are a beginner, I recommend you to read the site below:

https://bitcoin.org/en/getting-started

I have started opening a account in a exchange to buy bitcoins and transfering for a wallet.

The most important thing for you to understand is the importance of the Private Key.

The second point for utilizing Bitcoin is understanding that the address (public key) is where you target where to send Bitcoins to, and where to receive Bitcoins.

Once you know that it is important to Secure and never share and be cautious where you import your private key, then you can gain Bitcoins and reliably use them throughout life.

Check out this website BitAddress.org and observe how it gives you the Private Key (secret) and Public Key (share)

There is also a video tutorial about it: Video about using BitAddress.org and importing the key into an online wallet.