I have bought a lot of bitcoins from Circle. They are very convenient. The drawback is that you give them direct access to your bank account.

I cannot recommend Coinbase because they track you and they will ban you if they don't like what you are doing with your bitcoins.

As for a wallet, Bitcoin Core is not a great choice. Before using it, you have to download the block chain (which can take days). And every time you start it up, it has to validate its database and sync the block chain (which takes several minutes).

I recommend Airbitz on your phone because it is very easy to use and very secure, and Electrum on your computer.