Keeping yourself safe online is important for everybody, but women and girls who find themselves under the control/supervision of an intimate partner have special security concerns.
Whether you are using your home computer, where your partner can log on and find out what you've been doing, or you have taken the first step to safety and are using someone else's computer or the computer at the library, you need to get in the habit of erasing your tracks before you leave any computer. Privacy Rights Clearinghouse has some resources to get you going in the right direction.
You also need to be aware of security while using or carrying your smartphone, which is actually a little computer. PRC will get you started. While some of the links are geared towards telemarketing, be sure to refer to the other sections on unwanted calls, caller ID and wiretapping, and geotag.
Be aware that nothing you can do can get ahead of a determined adversary. The only really safe computer or phone is one that your abuser does not know about and has no access to.
While you may not think your partner capable of hurting you, the most dangerous time for someone in a controlling abusive relationship is when they start thinking about leaving, and begin to act on those thoughts. Abusive people are very tuned in to the object of their "affection", and will be curious to say the least about where you go, who you see, and what you do on the computer if they suspect you might be leaving.
Please take precautions.