The specific ins and outs of how your Internet banking will work vary depending on which bank you're with. However, there are many things that banks do the same or very similar, so it is worth taking a second to learn about how Internet banking works in general as well as reading the information your bank sends you.
To begin with, you have to register for Internet banking, if you did not do it when you set up your account. This is generally a matter of simply walking into your bank or phoning them and saying "I'd like to sign up for Internet banking". They will then send you a series of letters with various PIN numbers and passwords (including including a physical security device with a numeric keypad), along with instructions on how to use them to access your Internet banking.
Once you're in, you should be presented with a list of your accounts (if you have more than one) or a list of your recent transactions. This allows you to quickly see the status of all your accounts and what has happened to them recently. From here, you can access pages where you can make various kinds of one-off payments, and set up or cancel regular payments.
To make a one-off payment, such as a bill payment or sending money to someone else's account, you will need the person or company's bank account number – some banks will have a list of utility bills already built in to the website. Simply type in this information on the payment screen, together with the amount, and click 'pay'. The money should reach the other account within 2-3 working days.
Setting up regular payments is a little more complicated. Again, you need the numbers and amounts, but you also need to know the start date, end date, and how often you want the payment to be made. Be careful of exactly when you set your payments to leave your account, as the bank will often charge you a large fee if there is no money in the account when the payment is scheduled to be made.