Today is sunday and I slept until 12'o clock with my cat. As  [or 'Unfortunately, because'] I woke up late I can't help my sister to move out.She is moving near [or 'She is moving close to'] her college.I think it will be hard to get used to the fact that she will not be at home, because we have lived together for 20 years. But [don't usually start a sentence with but] it is a good way to let her grow up.I hope she will learn lots [you can use more but it's better if you use it when you are comparing something e.g. 'I hope she will learn more from living on her own than at hom] from living on her own [or 'living by herself'].

If you're writing formally generally you don't start sentences with 'BECAUSE, BUT or AND' but I think it's fine in something like this but most probably easier to remember if you just stick to that rule all the time...

Today

Today is sunday and I slept in until 12' o clock with my cat. Unfortunately, (yours is actually okay, but I was taught never to start a sentence with because! But it makes sense :)) because I woke up late, so I can't help my sister to move out. She is will moving to live near to nearby her college. I think it's will be hard to accustom get accustomed to the fact that she is not at home, because we have used to lived together for 20 years. But it is a good way to let her grow up. I hope she will learn more from livinge with on her ownself.