Is it possible to transfer Bitcoin from one wallet to another wallet? For example, if I buy bitcoin from Blockchain, can I transfer it to Coinbase?

And will there be a transfer fee involved or is it free?

Thanks :)
Of course you can. You bitcoins are not stored in the the wallet which may be a web wallet, hardware wallet or paper wallet. They are stored on the bitcoin network (the blockchain). The wallet is just a reference.

You can transfer bitcoins from any wallet to any other wallet.