Your home is a more casual environment than the office, but you still need to remain professional. Review all of the policies of your company and follow them at home as well as you would follow them in the office.
The Internet is full of content which is not appropriate for the workplace. Viewing pornographic or obscene material in the workplace is often tantamount to sexual harassment. It also just isn’t a very nice thing to do. When you are working from home you may be tempted to view websites and other material you would never think of looking at in the office.
Using the computer provided by your company, or accessing the Internet through your company VPN, to look at inappropriate material in your home is seen by many companies as the same as doing it in your office. There are hundreds of stories out there of people getting fired for the history in their browser. It is best to err well on the side of caution. Your work computer should be used only for tasks directly related to your work. If you want to spend time accessing extracurricular material then you should purchase a personal computer.
Most companies have you sign a contract including a non-competition clause. This normally means that you can’t work for other computer companies without permission. This doesn’t change if you are working remotely. Nobody is watching you work, but you still need to let your company know what you are working on.
Be careful about intellectual property issues. Some companies take the stance that they own any code your write while working for them. Others are more permissive. You need to contact your company for permission before writing any software you don’t want them to own. That includes open source software.