Here is the basic idea of a hot/cold wallet system:

Cold wallet:
The cold storage is done on an air-gapped machine, that is a computer that is not connected to any network. The private keys can be stored on this machine or on a paper wallet etc. The transactions are signed on this machine, the signed transaction is then moved to an internet connected machine where it is then broadcast to the network.

Hot wallet:
Hot wallet is usually kept on an internet connected machine. A small amount of BTC is kept on this machine, enough for day-to-day use. When it gets low it is topped up from the cold wallet.

I would not recommend using 3rd party API's to manage your hot wallet as it carries extra unnecessary risk. If coinbase were to get hacked you'd lose your BTC. If your machine is compromised oauth or 2FA won't protect you so it provides no extra security.

Those API's are only useful if for some reason you are unable to run a client or maintaining one is unnecessary for your use.

I recommend you take a look at the Armory Bitcoin client - https://bitcointalk.org/index.php?board=97.0