This morning, my mother telephoned me, and I am very pleased.

I love my father,mother and brother.

Each week l will give them a call.一般现在时，呵呵。

I told them I had a very good day/ time, and that they youneed not worry ahout me.因为现在是转述早上打电话时说的话，所以用them, 跟前面的they对应。

I also hope that my friends to call their Mom and Dad , and do not let their parents worry ahout them. 记住，可以说I hope to do, 但是不能说 I hope sb to do 哦，要表达这个意思只能说I hope + 从句，或者说I wish sb to do...

This morning, my mother telephoned me, so I was very pleased.

Grammatically, your sentence is fine, but I would simply say "I love my family" for the sake of simplicity.

Each week, I give them a call.

I told them that I was doing very well and that they do not need to worry about me (at all).

I also hope that my friends call their parents so that the parents would stop worrying about their children.