I've banked and shopped online for decades - with never a problem. I find PayPal very secure and convenient for one off small purchases, like sacks of bird food from Ebay, as I don't have to give my details every time, it's stored and done automatically.
For larger purchases, I tend to use my credit card (paid back every month) for the extra consumer protection it gives. My boss once booked a holiday with her debit card, the firm went bust - and she lost her money. If only she'd used her credit card, she would have been refunded.
For smaller stuff and supermarket deliveries, I use my debit card, though. I look at the web address when making a payment, just to check for a secure site sign e.g. https (the s indicates secure) and a padlock in the browser.
Fraud is mainly attempted on fake websites and through phishing emails rather than general buying online.