I wrote it is my grandmother's birthday today.

I thought birthday was 생길?

I think it doesn't accept "MY grandmother's birthday" because you can call any older woman you're close to 할머니. So it's not actually your grandmother you're talking about, if it were you'd have to specify that with using 제 for example.

it's grandma's birthday today

What's the difference between 할머님 and 할머니 or 할아버지 and 할아버님?

I keep seeing many people frustrated with this lesson. The truth is that because Korean has different honorfic levels, you really need a grammar book or another program to understand what Duolingo is trying to teach you. Many times with the Korean lessons, Duolingo does not give the context. It is helpful to remember the topic you are studying and use a book like 'Korean Grammar in Use' or another website like 'howtolearnkorean' I use to struggle a lot with this app, because this app is not at a point yet to fully explain the differences in people speaking. I think it's also helpful to ask questions and hope that others can give a good answer.