"I am looking at an envelope that my daughter brought home from school as they are raising money for Lepra. If she takes the money to school in the envelope the ONLY payment options are: cash, cheque or postal order. "
Yes, and in 1996 you could buy music on Vinyl, CD or Compact Cassette. That didn't make the LP and cassette any less doomed. When I bought some tickets for a Queen concert in 1980, I sent that nice Mr Goldsmith a postal order: the postal order is now for practical purposes as dead as Freddie.
No one is suggesting killing cheques tomorrow morning, because right now there are things they're still useful for. However, citing cases where they're still used because that's the way it's always been proves nothing: they're going away. And that's because they're too easy to forge, too easy to alter and too expensive to process. They've got ten years.
"How secure is online banking? I am very wary, so don't use it "
That's because people tend to over-estimate the risks of things they're unfamiliar with, and under-estimate the risks of things they think they understand. It's a great deal more secure that (speaking from painful experience) having a bag stolen from a gym containing both your cheque book and your cheque card. What are the risks you perceive in online banking? Speaking professionally, I have few concerns. I wish they used tokens for login (some do, my bank doesn't) but beyond that I think it's well managed.
"How secure is using credit/debit cards on the internet? Amazon automatically store card details which really annoys me, so every time I have completed a transaction I go into My Account and delete the card details."
What threat are you worried about? Because by any rational assessment you're doing precisely the wrong thing. Instead of using the facility to make repeat purchases without re-inputting the CV2 (the three-digit code on the signature strip), which Amazon cannot legally store, you're typing the complete stack of credit card details in every time, including the CV2. Because Amazon can't store the CV2, there's provision in merchant agreements for repeat transactions - same card, same address - to proceed without the CV2. Change anything, they need the CV2 to be re-typed.
So in order to guard against the risk of someone being able to make an order with Amazon for something you didn't ask for and have it shipped to you, instead you're passing the crown jewels of your CV2 through your home computer every time you buy something. The few times you type that, the better.
The risk with Amazon's storing of the credentials is that someone who breaks into your Amazon account can order something and have it shipped to your address. It's a risk: there have been scams where people do that, and then wait by the house and sign for it when the van comes. But it's a crude and risky crime, unlikely to succeed for long, which requires the criminal to be on-site.
On the other hand, every time you type your CV2 code, you're exposing it to a key logger or other malware. Unless I'm missing something, that's both a bigger risk and one much more likely to end in acrimony with the bank.
All the things that worry me about typing credit card details are at my end. The less I have to do it, the better. Honest companies can't do anything with your stored details. Dishonest companies wouldn't delete the stuff when you type "delete". And without the CV2, what can they do anyway?