Thanks.
The easiest and most secure way is to create a transaction from your old wallet (bc.i) to your new wallet (bitcoin core).
I would like to use my Blockchain wallet as my main bitcoin core wallet, can I do this?
You would need to export the private keys from ALL of the addresses in your blockchain.info wallet, then you would need to import ALL of those private keys into your Bitcoin Core wallet.
After that you should stop using the blockchain.info wallet completely. Any transactions that you continue to make in the blockchain.info wallet can create a data inconsistency issue with your Bitcoin Core wallet.
After you import all addresses, I suggest you to delete all addresses from your Blockchain.info wallet. Before deleting, I highly recommend you to make sure you have imported all addresses.
See "How do I delete an address?" question in https://blockchain.info/wallet/wallet-faq.