I do all my banking online. 4 bank accounts, 3 savings accounts 2 credit cards/accounts and a share dealing account and although I've switched banks many times during the past 25 years there has never been a problem and never had my details compromised.
It is very easy to transfer money. The only advice I can give you is not to transfer a couple of days before your pension is due to be paid in. It only takes 7 working days and compared with years ago it is a breeze.
I understand your concerns about online banking as I also had them. However it was pointed out that I could check my balance each morning and would know immediately whether something untoward had happened eg hacking.
With a paper statement once a month, hacking via the banks computers could occur immediately after the statement had been issued and the customer would be none the wiser for weeks.
My local bank has closed , so now to do a transfer of money I have to go to the next big town . I don't trust on line ! So this morning I'm thinking of changing banks but am nervous in case things don't go smoothly . Has anyone changed banks recently ?