The Internet is a great source of information and email can be a great way to communicate with family and friends. Be aware that someone who has access to your computer may be able to see what sites you have visited, or read your email messages.
You might feel safe sitting at your computer while your abuser is not around, but keep in mind that he or she may have ways of tracking your activity. If your abuser knows their way around a computer it may be safer to simply use a computer out of the house - at a friends, library, school, internet cafe, etc.
There are steps you can take to make it more difficult for someone to track your activities. Below is a link to questions and answers for some of the ways you can cover your internet activities.