It sounds very natural, but unfortunately, the grammar is not correct. This is a common mistake made by native speakers too. You should always list the other person first, and since you are doing an action, you use "I" instead of "me."
So, this way is best: "My friend and I are going to throw a party this weekend."
If you have trouble figuring out whether to use "I" or "me," remember that "I" does an action, and "me" receives an action. So 我打他 translates as "I hit him," but 他打我 translates as "He hit me."
To give another example, if someone else threw a party for you and your friend, you could say, "She threw a party for my friend and me." Why "me"? Because you are receiving an action, not doing it.
Hope this helps. :)