Converting to Judaism isn't as simple as it is to convert to Christianity.
It will take years of study and effort.
That aside, the govt of Israel will not "offer" you a chance for a Birthright trip. You have to apply for it and demonstrate you meet the requirements. I don't see why you wouldn't qualify, but the Israeli govt is rather close minded about these things. (EDIT: Someone else in the thread has said Birthright Israel accepts converts explicitly, so you should be good here)
THAT aside, don't go to Israel, don't give them any money. What they're doing right now is absolutely monstrous and it's morally inexcusable to support them.