"My mother would like to buy a few things."
It depends, if it involves close relationship or ownership you don't need it, but by now we can just all simply expect duolingo to expect to have the 的 with everyone
If you're talking about a person in your family or a close friend, someone you have a close relationship with, you don't really refer to them using "的“.
de is more for when you need to refer to someone that is outside of your family or who you are not close to, if you are reffering to your mother you use the phrase "我妈妈“ de can be omitted
I've see it omitted for many other personal relations as well. Even sometimes inantimate things like "my job"
as to use 的 or not, i think they are the same (i'm Chinese). whether they are your families or your close friends or just strangers , it's okay to add 的，，or not .
Shouldn't it be "想要" not just "想“ for "would like to"?
Was marked wrong, but my understanding is it would be more correct.