Your bank is probably already trying to switch you to online banking (it saves them money if you switch to online banking because you visit the branch less often and hardly ever need to call them) and you will need to sign up few documents for your online bank account (call your bank to find out how you can do it).  Once you have your user ID and password, you can get into your account from any computer connected to the Internet.  (Related: Online shopping safety)

Once you are logged in, you will be able to do most things that you can do in a branch and a lot more since the data is shown in real time.  So you can see all deposits, withdrawals, checks cleared, etc. as they happen.  But that is not the biggest benefit of online banking.  The biggest benefit is the ability to pay your bills online.  What you do is that the very first time you want to pay a bill, you enter the payee information (name, address, account #, etc. from your bill) and then just enter the amount and the date that you want the check to go out.  You get an immediate confirmation.  After that it is the bank's headache to make sure that your payment is sent to the payee.  In most cases, the banks have existing relationships with other institutions (e.g. credit card companies, phone companies, etc.) and they simply do an electronic transfer.  In cases where the payee is a small company (for example, the guy who comes and fills our oil tank and another who services our irrigation system), the bank simply prints a check with my name and mails it to them.  You don't need to worry about any of this and the bank even does not charge you for the stamps (if it does, you should change banks and we will get into that below).  (Related:  Personal finance for empty nesters)

How to select a bank so that you can do your banking online?

• If your current bank is offering it, you must sign up but do not pay a single penny to them either as a service charge or for anything else.  The bank should also put the stamps for you and mail the checks for you at no cost (it is fairly standard practice now).  If your bank wants to charge you for online banking, it is time to look at other banks in your area and on the Internet.  Once you have that information you can even "negotiate" with your existing bank
• You can also bank with a true "online bank" (the ones that do not have a physical presence).  Take a look at this alternative as well because my nearest branch is now over 150 miles away and I have not visited a branch in over 5 years now (when I have to deposit a check, I just mail it to the bank).  I can use any ATM that I want but if you are really concerned about paying the fees that banks charge for letting you use their ATM, you can simply ask for cash-back when you pay for your groceries.

How to make the best of online banking?

• Pay all your bills through your online account rather than going to the websites of the institutions themselves.  If your mortgage or credit card companies want you to pay the bills on their websites, don't do it.  It will only make your life complex since you will have too many accounts to track and passwords to remember.
• Stop all your electronic fund transfers (EFT) or automatic deductions.  Once you sign up for these, you lose control.  With online banking, you are in control.
• Set up automatic payments for amounts that do not change (e.g. your mortgage payments), and when you get the bill just check to see if there is no change.  If so, nothing to do.  If there is a change, just go and edit the amount to be sent out.