So what I've done is I've gone to bitaddress.org. Generated an address and private key. Screenshotted the page and saved the screenshot as a file on my laptop. Is this now a secure address that I can send some btc to so long as I don't lose the private key file? As far as I can see the only risk is someone hacking my computer and stealing that file, the private key has never been typed, so no keylogging issue. Yes?
Bad move - using bitaddress.org (online) is risky enough but then also saving screenshots of your private keys on your laptop means that you are just going to have to "hope" that no-one else is going to find them.
Key logging is a major problem but it isn't the only one. The only way to really safely secure your bitcoins is to generate your keys *offline* (and best to do that with a computer that *never* connects to the internet again).
For non-techies I think that the Trezor (or some similar) device might be the best answer.