Is it normal to feel sad all the time? What can I do about it?

Listen to your mum. What you have described is just part of being a teenager. You don't want to socialise or go out with friends. Other teenagers don't want to be home with parents or family. Both are perfectly normal. But, if your sad feelings persist then maybe you should talk to someone about them. Perhaps your mum, or another trusted adult, your teacher or doctor.