It would appear that your timer is faulty and needs to be replaced.
Typically when a timer starts to wear out, over time, the plastic steps that are part of the timing sequence can wear down and cause the washer to malfunction.
If you can hear the time motor running but it doesn't ever go into the rinse cycle, then it is most likely the timer motor that is the issue.
If you want to get an idea of the cost for obtaining a replacement timer, you can go to www.searspartsdirect.com
and enter in your complete washer model number into their parts search and then select the "control" link to see a line drawing and parts list for your washer.
Since you didn't provide a model number for your washer, I'll give you the generic instructions for doing the replacement. Again, with the sears link above, you'll be able to see the location and mounting method of the timer on your washer.
This is an easy repair for anyone to undertake, just be sure to unplug the washer from the power outlet before you attempt it. First, remove the timer knob from the front of the washer simply by pulling it straight off. then with either a socket wrench or nut driver, remove the nut and lock washer that holds the timer onto the panel.
Now remove the screws on the backside of the control panel so you can get at the timer. remove any mounting screws/hardware that holds the timer on the panel. Once you've done that, you can just unplug the wiring harness from the faulty timer and plug it into the new one. Now just remount the timer, put the back panel cover back on and fasten the timer shaft with the lock washer and nut, put the timer knob back on and give it a try.
That should get your washer back working properly. I hope you find this Very Helpful and best regards!